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PB-111586-R HANDBOOK OF PIEZOELECTRIC CRYSTALS FOR RADIO EQUIPMENT DESIGNERS John P. Buchanan Philco Corporation October 1956 DISTRIBUTED BY: CLEARINGHOUSE FOR FFKRAL SOCNTIFC AND TECHNICAl INFORMATION U. 8. DEPARTMENT OF COMMERCE / NATIONAL BUREAU OF STANDARDS / INSTITUTE FOR APPLIED TECHNOLOGY REPORT selection aids Pinpointing R&D reports for industry Clearinghouse, Springfield, Va. 22151 U.S. GOVERNMENT RESEARCH AND DEVELOPMENT REPORTS (USGRDR)-- ■SEMI-MONTHLY JOURNAL ANNOUNCING R&D REPORTS. ANNUAL SUBSCRIPTION $30.00 ($37. SO FOREIGN MAILING). SINGLE COPY $3.00. U.S. GOVERNMENT RESEARCH AND DEVELOPMENT REPORTS INDEX-semi-monthly INDEX TO U.S. GOVERNMENT RESEARCH AND DEVELOPMENT REPORTS. ANNUAL SUBSCRIPTION $22.00 ($27.50 FOREIGN MAILING). SINGLE COPY $3.00. FAST ANNOUNCEMENT SERVICE — summaries of selected r&d reports compiled and MAILED BY SUBJECT CATEGORIES. ANNUAL SUBSCRIPTION $5.00, TWO YEARS: $9.00, AND THREE YEARS: $12.00. WRITE FOR AN APPLICATION FORM. 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GOVERNMENT PRINTING OFFICE : 1968 O • 328-773 NOTICE This is your copy of the latest edition of "Handbook of Piezo Electric Crystals for Radio Equipment Designers" prepared by Phllco for the Air Force as a WADC Technical Report. The Report WADC TR-56-156 covers material consid- ered useful in the design of crystal oscillators for electronic equipment. It will serve well in the hands of the electronic design engineers of your organization. The report applies to the application of "Military Type" crystal units in equipments designed for the Armed Services. The report strongly in- dorses the use of military type crystal units, however, does not discourage use of non-military types if a need exists. It does recommend that your organ- ization bring to the attention of the "Frequency Control Group" of the indivi- dual service organization needs for non-military types before the design application is frozen so that military types currently in development may in turn be offered for your consideration. This cooperation is strongly requested so as to effectively make use of the military types which have established sources of supply. In the event this action is not taken, the special, non- military, type crystal unit may not be procurable when production quantities are most needed. Your support and cooperation is appreciated. Sincerely yours OwHN B. RIPPERE Colonel, USAF Chief, Comm & Nav Laboratory Directorate of Development Wright Air Development Center PB-Ilise&ii WM)C TECHNICAL A8T1 A DOeUM£#t HO. Afr 1 ‘ .j ' 1 ■ ■ » •! •••; ‘V' ' ‘ .<,/’■■' '• •! ' v' ^ '- •,■> , .. ' \ ,*■'’* ■ ■■'‘. ■■■:•'.-* . ‘ • H^^>J£)BOqK • ‘ '1 V ' i '. ‘ . , -f ■'f'i'J'i ' *''' V'''' ‘ • U' ^ *• '-•' ‘ . ' ' »T /'^ vcnr a t ■t-f'i. ; ' ,'l: ',. : r- • ■ , • ■■ - -i '•■ '■'•■ .’v.' ■ ^ ' c jrVYLt/uK^ti. K Anio i i\L FOB [ERS^'^r ^V..: T1»s Hewrt SuperMdes TR 5^^ Jqlutt- Prn-BiMihimacnt-' ■ ■ v "{j^; \ i ■'' •■.■ . v- 4 ' ‘ ‘ -'“-V/'V • " S''-' ■ . ■' A A'--'"":- ■■ V ,".. .A.' :, • ..Fildeo;0»vp^*tioi| ' .%< ...y •" " f-'" '. - ' 1' ,- • • • * • V . .. • ... -.• .v'-. ’'■•.H-'. •;• • ■ • ;•• ■’ ^ ■ >*‘ ,;■/■ , OCTOBER J956 ^ . ^ ■- ■ '■ /. - . • ■-■ ■■' -• '.if' . ' : • ^ .•••.'^ :•• ... ■ ••■■ -'• A ■■• i ■, •' ) WADC TECHNICAL REPORT 56-156 AST I A DOCUMENT NO. AD 110448 HANDBOOK OF PIEZOELECTRIC CRYSTALS FOR RADIO EQUIPMENT DESIGNERS This Report Supersedes WADC TR 54-248, Dated December 1954 Jokn P. Buckanan Pkilco Corporation OCTOBER 1956 Communications and Navigation Laboratory Contract No. AF 33(616)-2453 ARDC PROJECT 4155, TASK No. 43033 W^ri^kt Air Development Center Air Researck an«l Development Command United States Air Force Wri^kt-Patterson Air Force Base, Okio McGregor C Werner Midwest Corp, , Otytoa, O. 2>37-1000-10-13-Sa FOREWORD This handbook was prepared by the Technical Publications Department, TechRep Division, Philco Corporation under Contract AF33 (616) -2453. Mr. F. W. Wojcicki served as project director, with Mr. M. W. Nachman assuming these duties during the final processing stage. Mr. J. P. Buchanan was project engineer and author. Credit for assembling the data on specific crystal units and holders in Sections II and III belongs to Mr. C. W. Henry. This Task No. 43033, titled Handbook of Piezoelectric Crystals for Radio Equipment Designers, ARDC Project 4155 was administered under Mr. V. J. Carpantier as chief of the Specialties Section, Communication Branch of the Communications and Navigation Laboratory, Wright Air Development Center. Appreciation is extended to Gentile Air Force Depot, Squier Sigrnal Lab- oratory, Armour Research Foundation at Illinois Institute of Technology, New York University College of Engineering, and to the many individuals and other organizations whose generous cooperation has proved so im- portant during the preparation of the handbook. In particular, the suc- cessful conclusion of the project is heavily indebted to the interest, admin- istrative assistance, and many valuable suggestions of Mr. E. H. Borgelt of the Frequency Control Group, Wright Air Development Center, and to Mr. R. A. Sykes and assistants at Bell Telephone Laboratories for freely giving of their time and knowledge in reviewing the text and contributing important corrective comments for improving the usefulness and accuracy of the text. COPYRIGHT CREDITS Many of the illustrations used in the general information section of the handbook are copyrighted by D. Van Nos- . trand Co., 250 Fourth Ave., New York, N. Y. and are iden- tified by an asterisk (*) after each title. Reproduction of any of these photographs is prohibited without the express permission of the copyright holder. COPIES Copies of this report may be obtained from the Office of Tech- nical Services, U. S. Department of Commerce, Washington, D. C. at a nominal cost. WADC TR 56-156 ABSTRACT A comprehensive manual of piezoelectric control of radio frequencies is offered. It is directed toward the design of oscillator circuits having opti- mum operating conditions when employing Military Standard crystal units. Included is a survey of the development of the piezoelectric crystal art; descriptions and characteristics of all crystal elements and mounting methods that have found commercial application ; a detailed study of the equivalent circuit characteristics of crystal units; analyses of basic piezo- electric oscillator principles and of the effects of changes in various circuit parameters, using the Pierce oscillator as a reference circuit ; analyses and recommended design procedures for all types of piezoelectric oscillator cir- cuits used, or tested for use, in USAF equipments; schematic diagrams and tables giving actual circuit parameters of all available nonclassified piezoelectric oscillators now being used in USAF equipments ; descriptions of all crystal units and crystal holders now being used in USAF equip- ments, containing references and schematics of circuits employing those crystal units recommended for equipments of new design; a brief discus- sion of crystal ovens and descriptions of ovens currently available for use with Military Standard crystal units ; and a comprehensive index to increase the utility of the handbook as a reference manual. Circuit analyses, deriva- tions of equations, and suggestions for design innovations whose sources are not directly acknowledged have originated with the author and so far as is known have not been specifically confirmed in practice. PUBLICATION REVIEW This report has been reviewed and is approved. FOR THE COMMANDER; Col onel , c^. RIPPERE USAF Chief, Comm & Nav Laboratory Directorate of Development Wright Air Development Center WADC TR 56-156 iii TABLE OF CONTENTS SECTION I— GENERAL INFORMATION 1 SECTION II— CRYSTAL UNITS 430 SECTION III— CRYSTAL HOLDERS 569 SECTION IV— CRYSTAL OVENS 601 APPENDIX I— ACKNOWLEDGMENTS 629 APPENDIX II— BIBLIOGRAPHY. ....... 635 APPENDIX III— LIST OF MANUFACTURERS ... 654 APPENDIX IV— RELATED SPECIFICATIONS, STANDARDS, PUBLICATIONS AND DRAWINGS. 658 APPENDIX V— DEFINITIONS OF ABBREVIATIONS AND SYMBOLS 662 INDEX 683 WADC TR 56-156 iv SECTION I— GENERAL INFORMATION TABLE OF CONTENTS Subject Page INTRODUCTION Purpose and Scope of Manual 1 Control of Radio Frequency 1 The Piezoelectric Effect 1 Development of Piezoelectric Devices 2 PHYSICAL CHARACTERISTICS OF PIEZOELECTRIC CRYSTALS Descriptions of Useful Piezoelectric Crystals 5 Theory of Piezoelectricity 12 Modes of Vibration 15 Orientation of Crystal Cuts 18 Piezoelectric Elements 20 STANDARD QUARTZ ELEMENTS Types of Cuts 21 The X Group 22 The Y Group 34 FABRICATION OF CRYSTAL UNITS Initial Inspection of Raw Quartz 45 Inspection for Optic Axis and Optical Twinning 46 Use of Conoscope for Exact Determination of Optic Axis 46 Sectioning the Stone 47 Determination of X Axis 49 Cutting X Block 50 Determination of Twinning 51 Preparation of Wafers 51 Preparation of Crystal Blanks 51 Methods of Mounting Crystal Blanks in Crystal Holders 52 Housing of Crystal Units 67 Aging of Crystal Units 67 ELECTRICAL PARAMETERS OF CRYSTAL UNITS Equivalent Circuit of Crystal Unit 70 Simplified Equivalent Circuit of Air-Gap Crystal Unit 71 Effect of R-F Leakage Resistance 71 Effect of Distributed Inductance 72 Effect of Distributed Capacitance 73 Effect of Distributed Resistance 74 WADC TR 56-156 TABLE OF CONTENTS— Confinued Subject Page ELECTRICAL PARAMETERS OF CRYSTAL UNITS (Cont) Rule-of-Thumb Equations for Estimating Parameters 74 Impedance Cliaracteristics versus Frequency 77 Resonant Frequency of Crystal Unit 78 Antiresonant Frequency of Crystal Unit 80 Impedance Curves of Crystal Unit 80 Parallel-Resonant Frequency, f,,, of Crystal Unit 81 Typical Operating Characteristics of Crystal Unit 85 Measurement of Crystal Parameters 86 Methods for Expressing the Relative Performance of Crystal Units 90 Activity Quality of Crystal Unit 91 Frequency Stabilization Quality of Crystal Unit 98 Bandwidth and Selectivity Parameter of Crystal Unit 102 Crystal Quality Factor, Q 103 Stability of Crystal Parameters 105 CRYSTAL OSCILLATORS Fundamental Principles of Oscillators 113 Fundamental Requirements of Stable Forced-Free Oscillations 113 Application of Fundamental Oscillator Principles in the Design of Electronic Oscillators 114 Phase Rotation in Vacuum-Tube Oscillators 116 Types of Crystal Oscillators 119 Parallel-Resonant Crystal Oscillators 119 The Pierce Oscillator 120 The Miller Oscillator 187 Two-Tube Parallel-Resonant Crystal Oscillator 218 Oscillators with Crystals Having Two Sets of Electrodes 222 Crystal and Magic-Eye Resonance Indicator 223 Series-Resonant Crystal Oscillators 225 Meacham Bridge Oscillator 226 Capacitance-Bridge Oscillators 236 The Butler Oscillator 244 Transformer-Coupled Oscillator 258 Grounded-Grid Oscillator 266 The Grounded-Plate Oscillator 274 Transitron Crystal Oscillator 276 Impedance-Inverting Crystal Oscillators 277 Grounded-Cathode Two-Stage Feedback Oscillator 280 Colpitts Oscillators Modified for Crystal Control 281 Crystal Calibration 286 Synthesizing Circuits 288 Transistor Oscillators 342 Packaged Crystal Oscillators 356 Factors Involved in Oscillator Limiting 370 • CROSS INDEX OF CRYSTAL-OSCILLATOR SUBJECTS 375 WADC TR 56-156 Section I Introduction SECTION I— GENERAL INFORMATION INTRODUCTION PURPOSE AND SCOPE OF MANUAL 1-1. The purpose of this manual is to provide the design and developmental engineer of military electronic equipment with a reference handbook containing background material, circuit theory, and components data related to the application of piezoelectric crystals for the control of radio frequencies. 1-2. This manual is composed of the following sections : I. GENERAL INFORMATION II. CRYSTAL UNITS III. CRYSTAL HOI.DERS IV. CRYSTAL OVENS V. APPENDIXES 1-3. Section I contains a brief historical account of the discovery of the piezoelectric effect and of the application of crystal resonators as frequency- control devices, discussions covering the theory and physical properties of piezoelectric crystals, descriptions and performance characteristics of the more important quartz crystal elements, gen- eral discussions of the various crystal-unit fabri- cation processes and types of mounting, detailed discussions of the equivalent electrical parameters and performance characteristics of crystal units, and comprehensive qualitative and mathematical analyses of the various types of crystal oscillators, summarized with recommended design procedures. 1-4. Sections II, III, and IV provide the technical and logistical data, and information concerning the application of the crystal units, crystal holders, and crystal ovens currently recommended for use in equipments of new design. 1-5. The Appendixes contain the acknowledg- ments; a bibliography; a list of manufacturers associated with the piezoelectric crystal industry; a list of related U. S. Government specifications, standards, and publications ; a table of definitions for the abbreviations and symbols used in the Handbook ; conversion charts ; and an alphabetical index. CONTROL OF RADIO FREQUENCY 1-6. The greatly increased demand for military radio channels, with the consequent crowding of the radio-frequency spectrum, is, in the final analysis, a problem for the design engineer of fre- quency-control circuits. The problem is essentially one of providing a maximum frequency stability of the carrier at the transmitting station, and a maximum rejection of all but the desired channel at the receiving station. In each instance optimum results are obtained by the use of electromechani- cal resonators — maximum carrier stability is achieved by the use of crystal master oscillators, and maximum receiver selectivity is achieved by the use of crystal heterodyne oscillators and crys- tal band-pass filters. 1-7. The design of a constant-frequency generator has been an ideal of radio engineers almost from the beginning of radio science. Although many purely electrical oscillators have been devised which closely approach the ideal, none surpass the performance of the high-quality circuits employ- ing mechanical oscillators. Temperature-controlled oscillators having a sonic-frequency tuning fork as the frequency controlling element and followed by a number of frequency multiplying stages were the first of the radio-frequency generators employ- ing the high precision of mechanical control. The cumbersomeness and expense of the many multi- plier stages, however, have made the tuning fork oscillators impracticable insofar as the control of any but sonic frequencies are concerned. Today, precision control of radio frequencies has been made possible through the development of piezo- electric resonators, where the frequency-control- ling elements, usually quartz plates, have normal vibrations in the radio-frequency range. THE FIEZOELECTRIC EFFECT 1-8. The word piezoelectricity (the first two syl- lables are pronounced pie-ee') means “pressure- electricity,” the prefix piezo- being derived from the Greek word piezein, meaning “to press.” 1-9. “Piezoelectricity” was first suggested in 1881 by Hankel as a name for the phenomenon by which certain crystals exhibit electrical polarity when subjected to mechanical pressure. 1-10. That such a phenomenon probably existed seems to have been suggested first by Coulomb in WADC TR 54-248 1 QUARTZ CHARACTERISTICS S«clien I Introduction the latter part of the 18th century. His suggestions prompted Haiiy, and later A. C. Bequerel, into undertaking a series of experiments to see if elec- tric effects could be produced purely by mechanical pressure. Although both Haiiy and Bequerel re- ported positive results, there is some doubt as to whether these were not due to contact potentials rather than to piezoelectric properties of the sub- stances investigated — particularly since electrical polarities were reported in crystals that are now known to be non-piezoelectric. 1-11. It is to the Curie brothers, Jacques and Pierre, that the honor goes for having been the first (in 1880) to verify the existence of the piezo- electric effect. (For the initial report of their discovery, see paragraph 1-56.) 1-12. The Curie brothers tested a number of crys- tals by cutting them into small plates that were then fitted with tin-foil electrodes for connection to an electrometer. When subjected to mechanical pressure, several of the crystals caused the leaves of the electrometer to be deflected. Among those crystals showing electrical polarities were quartz, tourmaline, Rochelle salt, and cane sugar. In the year following these experiments, a prediction by Lippmann that the effect would prove reversible prompted the Curies to further investigations. The results verified Lippmann’s prediction by revealing that the application of electric potentials across a piezoelectric crystal would cause deformations in the crystal which would change in sign with a change in electric polarity. Furthermore, it was found that the piezoelectric constant of propor- tionality between the electrical and mechanical variables was the same for both the direct (pres- sure-to-electric) and the converse effects. In other words, the same polarization at the surface of the electrodes that results from a particular deforma- tion of the crystal can, in turn, if applied from an external source, produce the deformation. 1-13. It should be mentioned that the piezoelectric effect, which occurs only in certain asymmetrical crystals, is not to be confused with electrostric- ture, a property common to all dielectrics. Al- though electrostricture is a deformation of a di- electric produced by electric stress, it is unlike the converse piezoelectric effect in that its magnitude varies, not linearly with the electric field, but with the square of the field, and is unaffected by a change in the applied polarity. Electrostricture is the type of deformation a capacitor undergoes on being charged. In piezoelectric crystals this effect is normally small compared with the piezo- electric pi operties. DEVELOPMENT OP PIEZOELECTRIC DEVICES 1-14. From the time of its discovery until World War I, the piezoelectric effect found few practical uses. Those applications it did find appeared in the form of occasional laboratory devices for measur- ing pressure or electric charges. For the most part, however, little attention was attracted to piezo- electricity outside the crystallographer’s study. Nevertheless, during this time considerable theo- retical progress was made, due chiefly to the efforts of Lord Kelvin, Duhem, Pockels, and Woldmar Voigt. Voigt’s comprehensive Lehrhuch der Kristallphysik, published in 1910, is still con- sidered the reference authority on the mathe- matical relationships among crystal variables. 1-15. It was after the outbreak of World War I before serious attention was given to the practical application of piezoelectric crystals. During the war Professor Paul Langevin of France initiated experiments with the use of quartz crystal plates as underwater detectors and transmitters of acoustic waves. Although Langevin’s immediate purpose was to develop a submarine detecting de- vice, his research became of vital importance to many other developments. Not only did it attract the applied sciences to the possibilities of piezo- electric crystals, but also it initiated the modem science of ultrasonics. 1-16. The detecting apparatus that Langevin eventually devised employed quartz “sandwiches” which were coupled electrically to vacuum-tube circuits, and could be exposed under water where they would vibrate at the frequency of an applied voltage, or at the frequency of an incident acous- tic wave. The first function was employed to emit ultrasonic waves, and the second function to re- ceive and reconvert the echo into electrical energy for detection. 1-17. At the same time that Langevin was experi- menting with quartz as a supersonic emitter and detector, Dr. A. M. Nicolson, at Bell Telephone Laboratories, was independently investigating the use of Rochelle salt to perform the same functions at sonic frequencies. Indeed, his first application for a patent on a number of piezoelectric acoustic devices, April 1918, preceded by five months Lan- gevin’s initial application for a French patent. Employing Rochelle salt instead of quartz, because of its greater piezoelectric sensitivity, Nicolson constructed a number of microphones, loud- speakers, phonograph pickups, and the like. Among the circuits included in his 1918 patent application, was one that later proved of particu- lar interest — an oscillator employing a Rochelle WADC TR 56-156 2 Section I Introduction salt crystal as shown in figure 1-1. With this ex- ception, all the early applications of the piezoelec- tric crystal involved its use as a simple electro- mechanical transducer. That is, it was used either to transform mechanical energy in one system to electrical energy in another, or vice versa. Nicol- son’s oscillator was a distinct innovation in that it employed a piezoelectric crystal as a transformer of electrical energy to mechanical energy and back to electrical energy. 1-18. When Nicolson devised his oscillator, none of the possible functions of a piezoelectric vibrator had previously been investigated or discussed. His patent application offered no description of the crystal’s function, although presumably the crys- tal performed in some way to transfer part of the plate circuit energy to the grid circuit. Evidence that the normal vibrations of the crystal actually controlled the frequency seems to have existed, but no mention was made of this fact. The circuit, however, embodies the combined principles of coupler, filter, and resonator. Obviously the crystal acts as a coupler between the plate and grid cir- cuits ; and, inasmuch as the crystal may block the feedback of all plate energy except that at the frequency of the crystal’s normal mode of vibra- tion, the crystal may be imagined to perform the function of a filter, even though the over-all opera- tion is that of an oscillator. Finally, if the plate tap is connected at the bottom of the coil, so that the only feedback is through the plate-to-grid capacitance of the vacuum tube, the crystal may function as a conventional resonator, controlling the frequency as would a tuned grid tank circuit — the complete vacuum-tube circuit being the equiv- alent of a tuned-plate, tuned-grid oscillator. Thus, to Dr. Nicolson belongs the honor of being the first to employ the piezoelectric crystal purely as a circuit element, in all its principal circuit functions. 1-19. Although Nicolson was the father of the piezoelectric crystal circuit. Professor Walter G. Cady, of Wesleyan University, was its greatest prophet. In 1918 during a series of experiments being conducted to investigate the use of Rochelle salt plates for underwater signaling. Dr. Cady be- came interested in the electromechanical behavior of crystals vibrating in their normal modes. Out of the resonant properties that he discovered, he came to visualize the great possibilities that the piezoelectric crystal afforded as a resonator of high stability. After experimenting with several circuits, including the first quartz-controlled oscil- lator, Dr. Cady, in January 1920, not aware that Dr. Nicolson considered his oscillator controlled by the resonance of its crystal, submitted a patent application for the piezoelectric resonator, in which he reported its possibilities as a frequency stand- ard, filter, and coupler, and described the principles figura 1-1. The first crystal- (Rochelle salt) controlled oscillator. Invented by A. M. Nicolson, 1918 WADC TR 56-156 3 Section I Introduction of its operation. Although subsequent litigation concerning the discovery of the piezoelectric-reso- nator principle was decided in Dr. Nicolson’s favor, it is distinctly to Dr. Cady’s credit that he was the first to fully grasp the import of the piezoelectric resonator and to publish a public report of its prin- ciples and possibilities. His early pioneering in the field and his many later contributions have made Dr. Cady the American dean of piezoelectricity. 1-20. It soon became apparent that quartz crystals were the most stable and practical for use as reso- nators. Many investigators were attracted to the field, and progress was made both in the design and theory of crystal circuits. Professor G. W. Pierce of Harvard showed that quartz crystal os- cillators could be constructed with a single ampli- fier stage, as Nicolson had already done using Rochelle salt. This marked a considerable improve- ment over Cady’s oscillators, which had consisted of two or more vacuum-tube stages. Of particular note was the analysis by K. S. Van Dyke, in 1925, of the electrodynamic characteristics of a crystal resonator in terms of an equivalent electrical net- work : for the first time a way was opened to an understanding of the crystal resonator. In 1928, E. M. Terry showed that the frequency of a crystal oscillator was not entirely controlled by the crys- tal characteristics, but to a small degree was also dependent upon the other circuit constants. F. B. Llewellyn, in 1931, presented a classic analysis of oscillators showing the circuit impedance relation- ships that are necessary if the frequency is to be independent of variations in the voltage supply and vacuum-tube characteristics. Although the subject matter of this treatise deals with oscilla- tors in general, the principles are applicable to the design of crystal oscillators, if the electrical pa- rameters of the crystal are known. 1-21. The tuned-circuit oscillators of the early transmitters normally operated with heavy and variable loads. Many of the oscillators operated directly into an antenna, and in broadcast trans- mitters, modulation was performed in the oscilla- tor stage. This resulted in considerable frequency instability, and broadcast reception was often un- intelligible because of the frequency difference in radio waves arriving by different paths. It was in the determination of the cause and the correction of such interference that Messrs. R. Bown, D. K. Martin, and R. K. Potter of the Research and De- velopment Department of the American Telephone and Telegraph Company recommended the use of lightly loaded crystal-controlled oscillators fol- lowed by amplifiers. Under their supervision. Sta- tion WEAF in New York, in 1926, became the first crystal-controlled broadcasting station. 1-22. The principal factor limiting the stability of the early quartz oscillator was the relatively large frequency-temperature coefficient of the crystal, which allowed small changes in the ambient tem- perature to cause excessive changes in the reso- nant frequency. The immediate method of obtain- ing stability, of course, was to mount the crystal in an oven where the temperature could be con- trolled thermostatically. However, to decrease the temperature coefficient of the crystal, itself, also became the goal of a number of researchers. Be- cause some quartz plates exhibited positive tem- perature coefficients, whereas others exhibited negative coefficients, according to the orientation of the plate with respect to the axes of the mother crystal, the possibility arose that there should be some shape or median angle of cut which would have a zero coefficient. The first empiricists to en- ter the field were E. Giebe and A. Scheibe in Ger- many. In the United States, Mr. W. A. Marrison of Bell Telephone Laboratories turned his atten- tion to the problem of achieving the maximum precision possible in frequency control, and by 1929 had perfected a 100-kc frequency standard using a doughnut-shaped crystal (originally pio- neered by Giebe) with a nearly zero temperature coefficient. This success encouraged the Bell Lab- oratories research staff to launch a Concerted investigation into all phases of quartz crystal physics. Out of this program have arisen most of the principal advances in the design and produc- tion of quartz crystal units in the United States; although the early pioneering of S. A. Bokovoy and C. F. Baldwin at RCA has also been of notable significance. 1-23. Originally only the Curie, or X-cut, quartz plate was used — a plate in which the thickness dimension is parallel to the crystal’s X axis. Later the Y cut, where the thickness dimension is paral- lel to a Y axis, developed by E. D. Tillyer of the American Optical Co., began to compete with the Curie cut as the frequency-control element in com- mercial oscillators. By 1934, Messrs. F. R. Lack, G. W. Williard, and I. E. Fair of Bell Telephone Laboratories announced the discovery and devel- opment of two types of plates, called the AT and BT cuts, with such small temperature coefficients that they could operate stably under normal con- ditions without the use of temperature-controlled ovens. Concurrently, Bokovoy and Baldwin at RCA were experimenting with a series of crystals that they named the V cut, and their work, although of a less rigorous theoretical approach, substantially paralleled much of the research that was done at WADC TR 56-156 4 Bell Laboratories. In 1937, Messrs. G. W. Williard and S. C. Right announced the development of the CT, DT, ET, and FT cuts ; and by 1940 Mr. W. P. Mason had discovered the GT cut, the most stable resonator ever devised. The time-keeping stand- ards at both the Greenwich Observatory and the U. S. Bureau of Standards now use this crystal. Where other cuts exhibit a zero temperature co- efficient only at certain temperatures, the GT cat has almost a zero temperature coefficient over a range of 100°C. Besides the cuts discussed above, a number of others have been investigated which have proved particularly applicable for special uses. Among these are the AC, BC, MT, NT, 5- degree X, and the — 18-degree X cuts. 1-24. Paralleling the development of the new crys- tal cuts were the improvements made in the design of crystal holders. The early holders provided no means of “clamping” a crystal, for they were de- signed originally to accommodate X-cut plates whose favored modes of vibration required that the edges be free to move. Since the crystal in such a holder will slide about if used in equipment sub- ject to mechanical vibrations, a method of clamp- ing was needed before the crystal could be used in vehicular or airborne radio sets. Mr. G. M. Thurston of Bell Telephone Laboratories was led to the solution of this problem when he discovered that a crystal would not be restricted if clamped only at the mechanical nodes of its normal vibra- tions. The exact positions of these points, where the standing-wave amplitude is zero, depend, of course, on the particular mode of vibration. The low-frequency —18-degree X-cut crystal, for in- stance, can be held by knife-edged clamps running along its center, whereas AT- and BT-cut crystals can be clamped at their comers. Cantilever and wire supports which resonate at the crystal fre- quency have been devised for holding crystals at their centers. Although the mounting of crystals requires a far more exacting technique than for- Sacilon I Physical Characteristics of Piezoelectric Crystals merly, the crystal holder today provides support and protection sufficient to insure high perform- ance stability, even under the severe conditions of vibration that exist in military aircraft and tanks. 1-25. Unfortunately, the extremely critical nature of the design and production of crystal units has made it impracticable for manufacturers to mass- produce units with such exactitude that all the equivalent electrical parameters are standardized with an accuracy comparable to that now achieved in the case of vacuum tubes or other circuit com- ponents. However, definite progress has been made in this direction, and, if the need warrants the additional cost, reasonably exact characteris- tics may be obtained. For several years, each crystal unit had to be tested in a duplicate of the actual circuit in which it was to be used. This pro- cedure was disadvantageous from the points of view of both the radio design engineer and the crystal manufacturer. On the one hand, the radio engineer, knowing little more than the nominal frequency of the crystal unit to be installed in his circuit, could not achieve that degree of perfection in oscillator design which was otherwise theoret- ically possible. On the other hand, the task of making a given oscillator perform correctly effec- tively became the responsibility of the crystal manufacturer, since it was necessary for him to fit each crystal unit by trial and error to the par- ticular circuit for which it was intended. In recent years this cut-and-try procedure has been allevi- ated considerably by the development of standard test sets and by improvements in production tech- niques that permit more critical specifications. It is hoped that this handbook, by providing a more comprehensive description of the technical char- acteristics of the crystal units recommended for new design, will contribute in removing the limita- tions that too often in the past have forced the practical design engineer to approach his crystal circuits philosophically, rather than scientifically. PHYSICAL CHARACTERISTICS OF PIEZOELECTRIC CRYSTALS DESCRIPTIONS OF USEFUL PIEZOELECTRIC CRYSTALS 1-26. The piezoelectric effect is a property of a non-conducting solid having a crystal lattice that lacks a center of symmetry. Of the 32 classes of symmetry in crystals, 20 are theoretically piezo- electric, and the actual crystals which have been found in this category are numbered in the low hundreds. 1-27. Until the time of World War II only three crystals were commercially employed for their piezoelectric properties — quartz, Rochelle salt, and tourmaline. Today, the number is being in- creased by the development and application of synthetic crystals. Of these, the principal ones used in frequency selective circuits are ethylene diamine tartrate (EDT), dipotassium tartrate (DKT), and ammonium dihydrogen phosphate (ADP). See figure 1-2. WADC TR 56-156 5 Mction I Physical Characteristics of Piezoelectric Crystals Figur* 1-2. Commercially used piezoelectric crystals other than quarts WADC TR 56-156 6 1-28. Piezoelectricity is still in its infancy, and until more data have been collected and coordinated into a comprehensive atomic theory of the phe- nomenon, the chemist will have few clues to direct his search for a crystal having the maximum possible piezoelectric effect. Tourmalin* 1-29. Tourmaline is a semiprecious stone which at one time was called the “Ceylon Magnet.” This title seems to have been given it by early 18th century traders who introduced the stone to Europe, with the story of its strange magnetic property. If placed in hot ashes, tourmaline behaves as if it were electrified — first attracting ashes and then throwing them off. This is the phenomenon of pyroelectricity, closely associated with piezoelec- tricity, and was possibly the first electrical effect, other than lightning and St. Elmo’s fire, ever to be noticed by man. According to the theory proposed by Lord Kelvin, the pyroelectric effect of tourma- line is due to a permanently polarized lattice in the crystal, so that when heated, an unneutralized in- crease in the dipole moment occurs, proportional to the change in temperature and the coefficient of expansion. It was this pyroelectric theory of permanently polarized crystals that eventually prompted the Curie brothers to test for the piezo- electric effect. 1-30. Tourmaline is unsuitable for wide com- mercial use because of its expense and the scarcity in the number of large-sized natural crystals. Also, the temperature coefficients are negative for all tested modes of vibrations, which fact rules out the possibility of zero-coefficient cuts. 1-31. Tourmaline does have the advantage of dura- bility and a large thickness-frequency coefficient, so that for a given frequency it permits a more rugged crystal unit than quartz. For this reason it is sometimes used for the control of very high frequencies. However, the chief piezoelectric ap- plication of tourmaline is in the measuring of hydrostatic pressures. Rochell* Salt 1-32. Rochelle salt (NaKC 4 H 40 «- 4 HsO) is sodium potassium tartrate with four molecules of water of crystallization. The crystals are grown commer- cially by seeding saturated solutions of the salt and decreasing the temperature of the solutions a few tenths of a degree per day. They were first syn- thesized in 1672 by Pierre Seignette, an apothecary of La Rochelle, France, and until the time of Nicol- son’s inventions the salt was used primarily for its Section I Physical Characteristics of Piezoelectric Crystals medicinal value. Its exceptionally great piezoelec- tric effect — a blow with a hammer can generate as much as five thousand volts — has made Rochelle salt the principal crystal for use as a transducer in acoustic devices, such as microphones, loud- speakers, pickups, hearing aids, and the like. As a stable resonator it is far inferior to quartz, not only because of a greater sensitivity to tempera- ture variations, but also because of its tendency to disintegrate during extremes of humidity. If the ambient humidity drops below 35 per cent at room temperature, the water of crystallization will be- gin to evaporate, leaving a dehydrated powder on the crystal surface. Should the humidity rise above 85 per cent at room temperature, the salt will absorb moisture and begin to dissolve. For these reasons a Rochelle salt crystal should be mounted in a hermetically sealed container, or, if this is not possible, at least coated with wax. In the case of the former, if powders of both the crystalline and dehydrated forms of Rochelle salt are also enclosed within the sealed chamber, the humidity of the chamber will automatically increase or decrease with corresponding changes of temperature, and a stable balance between the crystal and chamber vapor pressures will be maintained. However, at a temperature of 55°C (130°F) the crystal, which is a double salt of tartaric acid, breaks down into sodium tartrate, potassium tartrate, and water. The solution formed will remain a viscous liquid for some time if super-cooled, and, as such, makes an effective glue for binding together plates of the crystal. 1-33. Although Rochelle salt, between the tem- peratures of --18°C and -|-24“C, has a greater piezoelectric effect than any other crystal, it seems that eventually it will be replaced by other syn- thetic crystals, in particular, ADP (NH 4 H 2 PO 4 ), which requires no water of crystallization. Never- theless, as an electromechanical transducer, Ro- chelle salt is still the most widely used of the piezoelectric crystals. ADP 1-34. ADP (NHiHjPO,), ammonium dihydrogen phosphate, was discovered and used during World War II as a substitute for Rochelle salt in under- water sound transducers. Like Rochelle salt, ADP crystals can be grown commercially; but unlike Rochelle salt, it requires no water of crystalliza- tion, and hence has no dehydration limitations, being able to stand temperatures up to 100°C (212°F). Also, ADP is more durable mechanically than Rochelle salt. 1-35. Although the crystal’s principal application WADC TR 56-156 7 Section I Physical Characteristics of Piezoelectric Crystals has been in submarine-detecting apparatus, its greater stability suggests the probability that it will eventually replace Rochelle salt as the prin- cipal transducer in other sonic devices. EDT 1-36. EDT (CeHijNzOs), ethylene diamine tar- trate, was discovered and developed during World War 11 as a substitute for quartz in low-frequency filter units. Quartz crystals at this time were in such great demand for the frequency control of military communication equipment, that a short- age developed in the supply of large-sized natural crystals which were needed for cutting filter plates of IV 2 to 2 inches in length. This shortage was acutely felt in the telephone industry, where there exists the chief demand for such plates for use in the band-pass filters of carrier systems. The dis- covery of EDT was the solution to this problem, for this crystal can be grown to any size desired, and it has the chemical stability (no water of crystallization), low mechanical loss, zero tem- perature coefficient, and small aging effects that make it a suitable substitute for quartz. 1-37. EDT is not as rugged mechanically, nor does it have quite as high a Q as quartz — although the EDT crystal units operating as filter elements in the 20- to 180-kc range do have Q’s in the neighbor- hood of 30,000. Moreover, for use as the frequency- control element in high-frequency oscillators, EDT is inferior to quartz because of its greater sensi- tivity to temperature changes. Even though high-frequency modes of vibration have been found with zero temperature coefficients, the temperature shift to either side of the optimum value must be kept approximately one-fifth that for a comparable quartz plate (BT cut, for example) in order to maintain the same frequency tolerance. Where only a minimum of temperature control might be needed for quartz, EDT will require fairly accurate control. Because of these disadvantages, EDT does not threaten at this time to replace quartz in high-frequency oscillators, but it does have promising possibilities for use in oscillators of the frequency-modulated type. Here, EDT plates have the advantage of a relatively wide gap between their resonant and antiresonant frequencies, thus permitting a large percentage swing of the oscillator frequency. If temperature- controlled, the EDT crystal can thus give crystal stability to a frequency -modulated transmitter. DKT 1-38. DKT (K.CiHjOo-'^HsO), dipotassium tar- trate, is another synthetic crystal which was in- vestigated at Bell Telephone Laboratories during World War II. The DKT molecule is similar chemi- cally to that of Rochelle salt except that the so- dium atom has been replaced by another potassium atom. The crystal, however, differs from Rochelle salt in that it contains only one molecule of water for each two DKT molecules, as compared with a water-to-salt molecular ratio of four-to-one in the Rochelle salt crystal, and it exhibits no tendency to dehydrate below 80°C (176°F). Also, the piezo- electric characteristics of DKT are less like those of Rochelle salt than of quartz. Indeed, in the lower-frequency filter circuits, DKT seems as promising as EDT as a substitute for quartz. 1-39. As compared with EDT, DKT has the advan- tage of better temperature-frequency character- istics. Zero temperature coefficients are possible where the frequency deviation on either side of the zero point is only one-third that for EDT. How- ever, DKT crystals are more difficult to grow than the EDT crystals, and primarily for this reason the development of a small EDT industry has already been established, whereas the DKT crystals are still in the laboratory stage. Raw Quarti 1-40. Quartz is silicon dioxide (SiOj) crystallized in hard, glass-like, six-sided prisms. The normal crystal structure is called alpha quartz; if the temperature is raised above 573°C (1063°F) most of the piezoelectric property is lost with a crystal transformation to beta quartz. At 1750°C (3182°F) the crystal structure is permanently lost, and the melted quartz assumes the fused amorphous form of silica. The density of alpha quartz at 20°C (68'’F) is 2.649 grams per cubic centimeter. The hardness of quartz is rated at 7 on Mohs’ scale — a greater hardness than glass or soft steel, but less than hard steel. 1-41. Silicon dioxide is believed to constitute ap- proximately one-tenth of the earth’s crust. It occurs in many crystalline forms such as quartz, flint, chalcedony, agate, onyx, etc., and in the fused amorphous state of silica, called “quartz glass.” Although quartz is an abundant mineral — sand and sandstone consist largely of quartz granules — large crystals of good quality are to be found in only a few areas. The chief source of supply has been Brazil, although large deposits of lower quality are also to be found in Madagascar and in the United States. Progress has been made in growing quartz crystals artificially. Such crystals are now commercially available, although this quartz source is still primarily in the develop- mental stage. The tremendous pressures required WADC TR 56-156 8 and the slow rate of growth have, until very re- cently, prevented quartz manufacture from being commercially feasible. Advances are now being made in growing imperfection-free quartz stones having major dimensions so oriented relative to the principal crystal axes that a desired type of quartz cut can be obtained with minimum waste. The future possibilities of quartz manufacture appear quite promising. 1-42. The large quartz crystals of geological origin are the products of long ages of growth under great pressure. The growing crystal assumes the shape of a hexagonal prism with each end pyramid- ing to a point. The prismatic faces are designated as m faces, see figure 1-4, and adjacent m faces always intersect at angles of 120 degrees. The opposite m faces of the prism are always parallel, but are rarely of the same dimensions. These faces are not perfectly planar, but are streaked with small horizontal growth lines, or striae. Parallel to the growth lines are the bases of the six end faces — three r and three z faces — which form a hexag- onal pyramid, but with only the r faces meeting at the apex. The end faces are quite smooth, with the r, or major, faces usually appearing more polished than the z, or minor, faces. Figure 1-3 shows a mother crystal with one of the pyramidal ends missing. Complete crystals are rarely found except in very small sizes. More likely both pyramidal ends will be missing, and frequently crystals are Figure 1-3. Raw <fuartx stone WADC TR 56-156 Section I Physical Characteristics of Piezoelectric Crystals found with all the natural faces broken or eroded away. The largest quartz crystal that has been re- corded was found in Brazil. It is described as a crystal of smoky quartz, 7 ft 2 in. long, 11 ft 2 in. in circumference, and weighing more than 5 tons. 1-43. Quartz is enantiomorphous — that is, it oc- curs in both right-handed and left-handed forms, which are mirror images of each other. The enantiomorphic faces of two ideal alpha-quartz crystals are represented in figure 1-4. The left- handed and right-handed forms are indicated by the direction in which the small upper x and s faces appear to be pointing. Note that this rule is valid regardless of which end of the crystal is turned up. However, the x and s faces are rarely found, so that the handedness of a crystal is usually determined by noting the optical effects when polarized light is passed through the crystal parallel to the optic (lengthwise) axis. IMPERFECTIONS IN QUARTZ 1-44. Pure quartz of structural perfection is a transparent, colorless crystal — such that the early Greek physicists believed it to be a perfected form of ice. Through the centuries quartz has been cut and ground into many ornaments, and was mysti- cally respected in the ancient art of crystal gazing. 1-45. The presence of impurities can convert quartz into a variety of gem-like colors. Amethyst, agate, and jasper are all quartz crystals colored by impurities. A different form of coloring is that which gives a smoky appearance to quartz. This effect differs in degree from crystal to crystal, and in extreme cases a crystal may be so dark that it cannot be inspected for defects nor for the align- ment of axes. However, by heating a smoky crystal from 350°C (662°F) to 500°C (932°F) it becomes t z Figure 1-4. Left and right quartz crystals 9 Section I Physical Characteristics of Piezoelectric Crystals quite as clear as the purest stone. Possibly the coloration is due to the dissociation of some of the silicon dioxide molecules, which recombine on heat- ing ; in any event, crystals which have been cleared of smokiness, remain clear, and have the same physical properties as the normal colorless crystals. 1-46. Other than those arising from chemical impurities, there are three types of structural defects to be avoided when cutting blanks from the raw quartz. These are cracks, inclusions, and twinning. Cracks 1-47. All raw crystals contain cracks to some extent, particularly near their surfaces, where fractures are easily caused by impacts. Tempera- ture variations and growth conditions are also causes of cracking. The larger cracks are readily visible, but not the separations with dimensions comparable to a wave-length of light. For this reason, any detected crevice should be assumed to extend somewhat beyond its visible length. Raw quartz should be handled with particular care, for the large crystals are more vulnerable to fractur- ing than are the small finished plates. No finished plate, however, should be permitted to contain a crack. Inclusions 1-48. Inclusions are small pockets, often sub- microscopic, holding foreign matter which was entrapped during the crystal’s period of growth. The trapped material may be a gas, liquid, solid, or any combination thereof. The pockets are often too small to be seen individually, but are readily detected by the shapes and coloring of the clusters they form. Groups of the smallest-size inclusions have a bluish cast; groups of medium-size in- clusions appear as a white frosting; and the larger inclusions are individually visible as small bubbles. Some of the clusters appear as small clouds; others appear as needles, which may be fine or feathery, and which may form parallel rows or spread comet-like from a bubble origin ; still other groups are draped in sheets or folds like veils; and, finally, there are those inclusions that arrange themselves in surfaces parallel to the natural crystal faces, outlining former growths, and appearing as crystal phantoms within a crystal. See figure 1-5. Not a great deal is known concerning the effect of in- clusions upon the performance of finished plates. However, the fine textured (blue) inclusions are the least objectionable, and the isolated bubbles are more to be tolerated than a veil or phantom. Blue needles are permissible in large, low-frequency plates that are not to be driven at high levels. Nevertheless, any inclusion weakens a crystal, and will not be present in a high-quality, finished plate. Twinning 1-49. Twinning is the intergrowth of two crystal regions having oppositely oriented axes. This ab- normality is rarely detectable by a casual visual inspection, and a crystal that appears homo- geneous throughout may, indeed, have several twinned areas; in fact, almost all large crystals have twinning to some extent. There are two types of twinning common to quartz — electrical twin- ing and optical twinning. In electrical twinning, only the electrical sense of the crystal axes is reversed, whereas in optical twinning, not only the electrical sense, but the handedness of the crystal Figure 1-5. Quartz crystal containing inclusions and fractures * WADC TR 56-156 10 structure is reversed — that is, one area will be right-handed and the other left-handed. 1-50. A finished plate, if it is to have predictable characteristics, must be cut entirely from a region having the same crystal structure; otherwise, the piezoelectric properties of one region will interfere with those of the other. Electrical twins are usually large, so both areas may be used separately for crystal blanks. Optical twinning, on the other hand, is usually confined to pockets, which are normally too small to provide crystal blanks, them- selves, so that only the predominant crystal region can be utilized. THE AXES OF QUARTZ 1-51. There are several crystallographic conven- tions by which the reference axes of crystals may be chosen, and much confusion has resulted in the past because of the various preferences of different crystallographers. Insofar as the over-all piezo- electric properties are concerned, the orientations of quartz have been universally measured accord- ing to rectangular sets of X, Y, and Z axes, with the XY, XZ, and YZ planes determined according to the crystal symmetries. However, even in this case, the choice of positive and negative axial and angular directions for right and left quartz re- mained more or less a matter of preference until the system proposed by the I.R.E. in 1949 became generally adopted. It is the I.R.E. system that will be followed here. It should be remarked first, how- ever, that a crystal axis is not intended necessarily to coincide with a central point in the crystal, but may represent any straight line parallel to the axial direction. It might also be noted that the different types of crystal faces are designated in this manual by the small letters m, r, s, x, and z, and these should not be confused with the capital letters X, Y, Z which denote the axes, nor with the small letters, x, y, z, when used to denote dimen- sions of a crystal in the axial directions. Z Axis 1-52. The Z axis is the lengthwise direction of the quartz prism and is perpendicular to the growth lines of all the m faces. It is an axis of three-fold symmetry, so that there are three sets of XY axes for each crystal (figure 1-6), with the direction of the Z axis common to all three. No piezoelectric effects are directly associated with the Z axis, and an electric field applied in this direction produces no piezoelectric deformation in the crystal, nor will a mechanical stress along the Z axis produce a difference of potential. Because the growth lines Section I Physical Characteristics of Piezoelectric Crystals are generally missing, optical effects are usually employed to locate the Z azis in raw quartz. (See paragraphs 1-121 to 1-124.) Quartz properties are such that light waves passing through a crystal are effectively divided into two rectilinear com- ponents, with one component traveling faster than the other except when the light ray is directed parallel to the Z axis. The optical effects are found to be symmetric about the Z axis, and thus whereas optical instruments may be used to determine this axis, they cannot be used to distinguish an X from a Y axis. For this reason the Z axis is commonly designated as the optic axis. The optical effects associated with the propagation of polarized light parallel to the optic axis not only are used to locate the Z axis in unfaced quartz (crystals, such as river quartz, whose natural faces have been de- stroyed), but to identify left from right quartz, and to locate twinned regions. Plane polarized light traveling parallel to the optic axis will be rotated in one direction or the other according to whether the crystal is left or right. To an observer looking toward the light source the rotation will be clock- wise for right-handed quartz and counterclockwise for left-handed quartz, with the amount of rota- tion depending upon the wavelength, being greater for blue light (short wavelength) and less for red light (long wavelength). Since the crystal lattice along the optic axis has no properties that distin- guish one direction from the other, the choice of the -1-Z and the — Z reference directions are en- tirely arbitrary for either right or left crystals. '^2 Figure 1-6. XY plane of quartz showing three sets of rectangular axes; X,Y,Z, X^Y^Z, X.,YjZ (Z axis is perpendicular to plane of paper) WADC TR 56-156 11 Section i Physical Characteristics of Piezoelectric Crystals Y Axis 1-53. The Y axes are chosen at right angles to the Z axis and to the growth lines of the m faces. See figure 1-4. For either left or right quartz, the positive end of a Y axis emerges from an m face that is adjoined by a z face at the end selected as the +Z direction. The Y axes are generally called the mechanical axes in contradistinction to the X axes, which are called the electrical axes. These names originated from the fact that simple com- pressional and tensional mechanical stresses along either an X axis or a Y axis would cause a polariza- tion of the X axis, but not of the Y axis. The names are somewhat misleading, for polarization in the Y direction is also possible if a crystal undergoes shearing or flexural strains. In practice, the Y axis of a quartz stone is usually determined after the Z and X axes have been located. X Axis 1-54. The X axes are parallel to the growth lines of the m faces, and to the lines bisecting the 120- degree prism angles. The positive end of an X axis is the direction that forms a right-handed coordi- nate system (see figure 1-4) with the Y and Z axes. This makes the directional sense of the X axis in right quartz the reverse, rather than the mirror image, of that in left quartz. Thus, in right quartz the negative ends of the X axes emerge from the prism corners that lie between the x faces, where- as, in left quartz the positive ends emerge from these .corners. 1-55. In either right or left quartz when the X axis undergoes a tensional strain (stretching), a positive charge appears at the end emerging be- tween the X faces; and when the X axis is com- pressed, this end becomes negatively charged. The X axis of raw quartz is usually determined by optical and x-ray methods. See paragraphs 1-126 and 1-127. THEORY OF PIEZOELECTRICITY Report Announcing Discovery of the Piezoelectric Effect 1-56. The theory of the cause of piezoelectricity stated in the most general terms is substantially the same today as it was at the time of its dis- covery. The following is the original report by Pierre Curie on the piezoelectric effect, which in- cludes a statement of the theory that led to its discovery. The paper was read at the April 8, 1880, meeting of the societe mineralogique de France, and is recorded in the Bulletin, soc. min. de France, volume 3, 1880. 1-57. “Crystals which have one or more axes whose ends are unlike, that is to say, hemihedral crystals with inclined faces, have a special physical property, that they exhibit two electric poles of opposite names at the ends of those axes when they undergo a change of temperature : this is the phe- nomenon known as pyroelectricity. 1-58. “We have found a new way to develop elec- tric polarization in crystals of this sort, which con- sists of subjecting them to different pressures along their hemihedral axes. 1-59. “The effects produced are analogous to those caused by heat: during a compression, the ends of the axis along which we are acting are charged with opposite electricities; when the crystal is brought back to the neutral state and the com- pression is relieved, the phenomenon occurs again, but with the signs reversed; the end which was positively charged by compression becomes nega- tive when the compression is removed and re- ciprocally. 1-60. “To make an experiment we cut two faces parallel to each other, and perpendicular to a hemi- hedral axis, in the substance which we wish to study ; we cover these faces with two sheets of tin which are insulated on their outer sides by two sheets of hard rubber; when the whole thing is placed between the jaws of a vise, for example, we can exert pressure on the two cut surfaces, that is to say, along the hemihedral axis itself. To per- ceive the electrification we used a Thomson elec- trometer. We may show the difference of potential between the ends by connecting each sheet of tin with two of the sectors of the instrument while the needle is charged with a known sort of elec- tricity. We may also recognize each of the elec- tricities separately; to do this we connect one of the tin sheets with the earth, the other with the needle, and we charge the two pairs of sectors from a battery. 1-61. “Although we have not yet undertaken the study of the laws of this phenomenon, we are able to say that the characteristics which it exhibits are identical with those of pyroelectricity, as they have been described by Gaugain in his beautiful work on tourmaline. 1-62. “We have made a comparative study of the two ways of developing electric polarization in a series of non-conducting substances, hemihedral with inclined faces, which includes almost all those which are known as pyroelectric. 1-63. “The action of heat has been studied by the process indicated by M. Friedel, a process which is very convenient. WADC TR 56-156 12 1-64. “Our experiments have been made on blende, sodium chlorate, boracite, tourmaline, quartz, cala- mine, topaz, tartaric acid (right handed), sugar, and Seignette’s salt. 1-65. “In all these crystals the effects produced by compression are in the same sense as those pro- duced by cooling; those which result from reliev- ing the pressure are in the same sense as those which come from heating. 1-66. “There is here an evident relation which allows us to refer the phenomena in both cases to the same cause and to bring them under the fol- lowing statement : 1-67. “Whatever may be the determining cause, whenever a hemihedral crystal with inclined faces, which is also a non-conductor, contracts, electric poles are formed in a certain sense ; whenever the crystal expands, the electricities are separated in the opposite sense. 1-68. “If this way of looking at the matter is cor- rect, the effects arising from compression ought to be in the same sense as those resulting from heating in a substance which has a negative co- efficient of dilation along the hemihedral axis.’’ Asymmetrical Displacement of Ciiar9e 1-69. The atomic lattice of piezoelectric crystals is assumed to consist of rows of alternating centers of positive and negative charges so arranged that the structure as a whole has no center of sym- metry. When such a lattice undergoes a deforma- tion, a displacement will result between the “centers of gravity’’ of the positive and negative charges. It is this displacement that results in a net unneutralized dipole moment, the polarity of ^ ® © 1 ® 4 - -f ^ 1 ® © j i POPLARITY UNDER NEUTRALIZED POLARITY UNDER COMPRESSIONAL DIPOLE TENSlONAL • STRAIN STRAIN Figure 1-7. Effective polarities resulting from sudden displacements of the centers of charge of a neutral- ized dipole. (If after displacement, the crystal were maintained indefinitely in the strained position, the effective polarity would eventually be neutralized by an accumulation of ions at the poles. A sudden return from such a state would thus result in an effective polarization in the unstrained position! Section I Physical Characteristics of Piezoelectric Crystals which depends upon the previous equilibrium posi- tions of the positive and negative centers of charge and the direction of the displacement, as indicated in figure 1-7. 1-70. In the case of a crystal with a center of sym- metry, a uniform strain in the crystal will always result in as much displacement of like charges in one direction as in another, and hence there will be no net shift of the centers of opposite charge relative to each other. A distribution of charges having a center of symmetry is illustrated in figure 1-8. Note that the centers of charge, both positive and negative, are at the geometrical center. If a uniform stress — compressional, or shearing — is applied along any axis, it can be seen that the center of either type of charge will at all times remain undisturbed, and thus the net piezoelectric effect will be null. 1-71. Lord Kelvin was the first to propose a molec- ular model with a charge distribution designed to explain the physical and electrical characteristics of alpha quartz. See figure 1-9. This model was accepted generally by the crystallographers until the theory failed to conform to X-ray tests. When a beam of X-rays enters a crystal, the intersecting atomic planes can be likened to partialh’^ silvered mirrors, each passing part of the beam, but reflect- ing the rest. Since the distance between adjacent parallel planes is on the order of an X-ray wave- length, the waves reflected from adjacent planes will tend to alternately annul and reinforce each other as the angles of incidence vary. The inter- ference pattern oh a photographic film will show an array of spots indicating the angles at which the reflected waves from different planes arrive in phase. From such data, with the X-ray wave- length known, it is possible to determine the rela- tive orientation of, atomic planes, and hence to reconstruct the arrangement of the atoms in the crystal. The X-ray data on alpha quartz reveals a 1 %) 1 % 1 Figure 1-8. Example of distribution of charges having a cenfer of symmetry * WAOC TR 56-156 13 Section i Physical Characteristics of Piezoelectric Crystals \ _ /K _ / \ @-G) ®-Q STQ. / \ / \ / \ / N / /®~®\ / \ / ^ ©-© ©-©^ ' ^ \ ,©-a \ ^©-©^ Vtn y^' 'y->w y— v' ©-© / \ ®— © / \ Figure 1-9. Kelvin's molecular model of the charge distribution of alpha quartz. The positive direction shown for the X axis corresponds to that of right quartz, for a compression of the crystal along that axis will cause the piezoelectric polarities to coincide in sign with the X-axial directions. For left quartz, the sign of the Y, as well as the X, axis, must be reversed in order to maintain a right-handed coordinate system * ARRANGEMENT OF ATOMS IN ALPHA QUARTZ, VIEWED ALONG AN X AXIS Figure 1-10> Arrangement of atoms In alpha quartz. Plane of paper corresponds to YZ plane In crystal Figure 1-11. Equivalent distribution of charges that account for observed piezoelectric effects of alpha quartz. (A) Piezoelectric polarity along X axis of right quartz due to compression along Y axis. (8) Piezoelectric polarity of Y axis of right quartz due to shearing stress, where the resultant strain is equivalent to a compression along the axis designated GH. (Note that in both A and 8, the piezoelectric effect is due to a rocking of the axial dipoles, and not to their compression or extension. To achieve the same deformations by the converse effect, equal voltages, but opposite in sign to the polarizations indicated, are applied across the respective axes)* WADC TR 56-156 14 more complex structure than was once suspected. See figure 1-10. Nevertheless, the early crystal model, as postulated by Lord Kelvin, still can be accepted as an approximation if we treat a single one of his molecules as representing simply the equivalent charge distribution within the lattice, as indicated in figure 1-11. 1-72. Figure 1-11 A shows the displacement occur- ring when a compressional stress is applied along the Y axis, or a tensional stress is applied along the X axis of a right-handed crystal. Note that the center of positive charge shifts in the negative direction of the X axis, and that the center of nega- tive charge shifts in the positive direction; how- ever, there is no net displacement along the Y axis. If the direction of the stress is reversed, so also is the effective polarity. 1-73. The polarization of the Y axis due to a shear- ing strain is illustrated in figure 1-llB. Assume that vectors A and B represent a simple shearing stress applied at right angles to the Y axis. If A and B are equal and opposite forces, there will be no displacement of the center of mass; however, since these forces are not directed in the same straight line, they create a couple which would maintain a rotational acceleration about the center of mass unless opposed by an equal and opposite couple. This counter-couple is represented by vec- tors C and D. If now, the forces are combined vectorially, they may be represented as a longi- tudinal tension in the EF direction, or as a longi- tudinal compression in the GH direction. Consider the charge displacement from the point of view of a GH compression. Note that each of the positive charges is forced to shift slightly in the -1-Y direc- tion, whereas each of the negative charges is dis- placed in the — Y direction. The net separation of the centers of charge thus causes the Y axis to become positively polarized at its geometrically positive end, and negatively polarized at its geo- metrically negative end. 1-74. Since the compression can be further ana- lyzed into two components of equal magnitude — one horizontal, and the other vertical — it can be seen (figure 1-llA) that the polarities which these would induce along the X axis tend to cancel (for reinforcement to occur, one of the rectangular components would need to be tensional and the other compressional) , and hence little or no polari- zation will appear in this direction. MODES OF VIBRATION 1-75. If a piezoelectric crystal is suddenly released from a strained position, the inertia and elasticity Section I Physical Characteristics of Piezoelectric Crystals of the crystal will tend to maintain a state of mechanical oscillation of constant frequency about one or more nodal points, lines, or planes of equi- librium, and alternating voltages will appear ac- cording to the particular mode of vibration. These are called the normal, or free, vibrations of a crystal, as distinct from the forced vibrations due to applied alternating mechanical or electrical forces that may differ in frequency from the crys- tal’s natural resonance. The normal vibrations may, in turn, be of two general types; the free- free and the clamped-free vibrations. Free-free vibrations ai-e those which would occur if a vibrat- ing crystal were floating in empty space, where, regardless of the particular mode, the center of gravity is a nodal point. Clamped-free vibrations are those that would occur if a crystal Were clamped at some point, or points, thereby prevent- ing all normal modes except those at which nodes occur at the clamped points. For example, in a free- free vibration the ends of the crystal are free to move; however, if these ends are clamped, the resonant vibrations must be such that the ends become nodes. However, if a crystal is clamped only at those points which would be nodes in a free-free vibration, in the ideal case no interfer- ence results, and the resonance is still that of a free-free mode. 1-76. There are three general modes of vibration for which quartz crystal units are commercially designed: extensional, shear, and flexure. Funda- mental vibrations of each of these modes are illus- trated in figure 1-12. Higher harmonics up to and including the fifth are also widely used. Harmonic vibrations higher than the seventh have special high-frequency applications, but are rarely em- ployed commercially. 1-77. A variation of the shear vibration is the torsional mode, which is readily excited in cylin- drical crystals; however, except for laboratory T^i ( B ) I D> Figure 1-12. Useful fundamental modes of quartz plates. (A) Flexural. (B) Extensional or longitudinal. (C) Face (or length-width) shear. (D) Thickness shear.* WADC TR 56-156 15 Sactien i Physical Characteristics of Piezoeiactric Crystals measurements of the properties of solids and liquids this mode is not in general use. Frequency of Quarts Vibrations 1-78. The frequencies of the normal mechanical oscillations of a quartz plate may be considered as those at which standing waves will be established by reflection from the crystal boundaries. The posi- tions of the nodes of the standing waves are pre- determined by the geometry of the crystal, and by any difference that may exist in the velocities of propagation for the different wave components. The wavelength of a particular mode (but not the wave shape, if the velocity of one component dif- fers from that of another) conforms only to the dimensions of the crystal faces. The frequency is related to the wavelength by the equation : f = I 1-78 (1) where: v = velocity of propagation X = wavelength The fundamental equation of the velocity of propa- gation is: 1—78 (2) where: c = stiffness factor in the direction of propa- gation p = density where: s = - = elastic compliance factor in the di- ^ rection of propagation Length- (or Width-) Extensienal Mode 1-79. The motion of the atoms in an extensional mode is parallel to the direction of propagation. In the case of rectangular plates, stationary waves are established in the length direction by the inter- ference of reflections from the opposite ends, where the wavelength is given by the formula : X = n 1—79 (1) where 1 is the length and n is an integer (1, 2, 3, etc.) equal to the harmonic. Thus, the frequency of a length-extensional mode is : f = ^ 1-79 (2) or, as expressed in terms of a frequency constant: f = ^ 1—79 (3) where: kj = ^ = frequency constant for length-extensional mode This formula, as well as the similar formulas for the shear and flexure modes, can be used to indi- cate the approximate dimensions required for a particular frequency when the appropriate fre- quency constant is known. Although the velocity of propagation decreases somewhat as the fre- quency increases, because of an increase in the frictional losses, this decrease is negligible for most purposes, and the same frequency constants that hold for the fundamental are also valid for the first few overtones. However, because of the coupling that exists between the length-extensional mode and other modes, the effective value of k will vary with changes in the w/1 (width/length) ratio. Equation (3) also applies to width-exten- sional modes except that 1 is replaced by the width, w. Thlcknasi-Extensionai Mod* 1-80. This mode is little used today because of the close coupling that exists between it and the over- tones of other modes. It is a mode that can be excited in a crystal whose thickness dimension is parallel to the electrical (X) axis (X-cut crystal) — the vibrations being such that the crystal alter- nately becomes thicker and thinner. Formerly, when X-cut crystals were widely used, the same crystal was often employed for the control of either a high- or a low-frequency circuit — using the thickness-extensional mode for the former and the length-extensional mode for the latter. Today, however, the more stable thickness-shear mode has almost entirely replaced the thickness-extensional mode in high-frequency circuits. The thickness- extensional frequency is given by the formula: nv _ nkj 2t “ ~r 1—80 (1) where v is the velocity of propagation in the thick- ness direction, n is the harmonic (n = 1, 3, 6,— for practical cases, although even harmonics of very small intensities have been observed), and kj is WADC TR 56-156 16 the generalized frequency constant. Actually, the effective thickness, t, decreases somewhat for the overtones, so that the correct value of kj is slightly greater for the harmonics than for the funda- mental. Thickaest-Shear Mode 1-81. The motion of the atoms in a thickness-shear mode is parallel to the major (length-width) faces of the crystal, whereas the wave propagation is parallel to the thickness dimension. The equation for the fundamental frequency when the thickness is very small compared with the length and width is: f = J 1-81 (1) where: v = velocity of propagation along thickness dimension t = thickness or, as expressed in terms of a thickness-shear fre- quency constant: f = ^ 1—81 (2) The thickness-shear is also called the “high- frequency shear” in contradistinction to the “face,” “length-width,” or “low-frequency” shear. The overtones of the thickness-shear mode may have components (reversals of phase) in the length and width directions as well as along the thickness. The more general formula for the frequency is: where m, n, and p are integers representing the harmonic component in the t, 1, and w directions, respectively. The above equation applies to an iso- tropic medium; however, since the elastic con- stants in quartz are not the same in all directions, the thickness-shear formula has been modified to : f ~ 1^3 W d" ni j3 "t" n*' (P - D* w* 1-81 (4) where ai and a, are constants to be determined empirically. For most applications, t is much smaller than 1 and w, and n = p = 1, so that the formula, f = sufficiently accurate. Fac»>Shcar Med* 1-82. The face-shear mode involves a more com- WAOC TR 56-156 17 Section I Physical Characteristics of Piezoelectric Crystals plex relation among the crystal dimensions. A complication arises from the fact that the wave is effectively divided into two components— one propagated along the length, and the other along the width. Each of these separate components has its own series of possible harmonics, so that the resultant frequencies of the face-shear modes are not necessarily integral multiples of the funda- mental. The approximate-frequency equation is : ^2 „2 m n -p-+ ki m^ + ar w^ 1—82 (1) where m and n are integers representing the length and width harmonics, respectively. The symbol a, is a constant of proportionality, approximately equal to one, which is inserted when the velocity of propagation along w is not the same as that along 1. If the face of the plate is square, the formula for the fundamental frequency is reduced to approximately: f = — 1—82 (2) w where: Ic* = k^-\/2 The fundamental vibration, where m = n = 1, is shown in figure 1-12C. Note that the shape of the deformation is not that of a parallelogram, as it would be if the plate were slowly compressed along a diagonal. Rather, the vibrational distortion is a dynamic one, and the resultant wave must be in the same phase at all points. Figure 1-13 repre- sents the face-shear mode for m = 6, n =; 3. Note that the number of nodes in each row is equal to m, and the number in each column is equal to n. figurm 1-13. faco-ehaar modi* for m = 6, n = 3. Dots inditato itodot * Section I Physical CharactorisHcs of Piezooloctric Crystals Longth-Width-Flexnral Mod* 1-83. The length-width-flexural mode is a bending of the crystal in the length-width plane. Normally, the crystal is so mounted that the ends are free to vibrate in a free-free mode. The formula for the frequency involves the root of a transcendental equation, but expressed in tenns of a frequency constant, the equation becomes: f = ^ 1-83 (1) The convenience of a common frequency constant for all practicable harmonics is not realized in the case of length-width flexures, where the “constant” ks is a function not only of the particular harmonic, but also of 1 and w. However, for long, thin rods ^^less than 0.1, where n is the harmonic) k, is approximately independent of the dimensions, and fixed values of k, can be assumed for the particular harmonics of different types of cuts. Because of the elastic cross constants in quartz, which relate a field in one direction to a polarization in a per- pendicular direction, a flexure may be accompanied by a torsion. To prevent this, the length of a crys- tal to be operated in a flexural mode should lie somewhere in a YZ plane. Length-Thickncts-Flexural Med* 1-84. Length-thickness flexures are used to control frequencies in the audio range. To obtain this mode, two long, thin plates of the same cut are cemented together with the electrical axes opposed, so that, when an alternating voltage is applied across the outer faces, one crystal strip expands as the other contracts, and vice versa — the over-all effect being a flexural vibration. The normal fre- quency of a free-free length-thickness flexure is given by an equation similar to that for the length- width flexure, except that the thickness, t, is sub- stituted for the width, w. Thus: f = nka (p) 1-84 (1) Frequency Ranqe of Normal Modes 1-85. Standard quartz crystal units are designed for frequencies from 400 cycles to 125 megacycles per second. Laboratory devices have employed thickness flexure crystals for the control of fre- quencies as low as 50 cycles per second, and, by exciting the higher thickness-shear modes, control of frequencies higher than 200 megacycles per second have been realized. At these high frequen- WADC TR 56-156 18 cies, however, so many interlocking modes are possible that it is difficult to prevent a crystal from jumping from one mode to another during slight variations of temperature, unless a very precise fabrication of the crystal unit has been achieved. The high-frequency limit of the lower harmonics is reached when the dimensions are so small that either the crystal cannot be driven without the risk of shattering, or that the impedances intro- duced by the mounting become proportionately too large for practicable operation. 1-86. The practical frequency ranges of the differ- ent modes are as follows : Flexure Mode — Length-thickness : 0.4 to 10 kc Length-width: 10 to 100 kc Extensional Mode — Length : 40 to 850 kc Thickness : 500 to 15,000 kc Shear Mode — Face : 100 to 1800 kc Thickness (fundamental) : 500 to 20,00Q kc Thickness (overtones) : 16,000 to 125,000 kc ORIENTATION OF CRYSTAL CUTS Riqht-Haiidad Caardinatu Sysfam 1-87. With the positive sense of the quartz X, Y, and Z axes determined as in paragraphs 1-52, 1-53, and 1-64, the positive sense of rotation about the axes is fixed by the conventions of a right-handed z Figure I -14. Positive directions of angles of rotation according to conventions of right-handed coordinate system coordinate system for both right and left quartz. If one imagines a' right-handed screw pointing towards the positive end of an axis of rotation, as represented in figure 1-14, the direction of an angle of rotation is considered positive if the rotation advances the screw in a positive direction — ^this corresponds to a clockwise rotation if observed when looking towards the positive end of the axis of rotation. The reverse, or counterclockwise, angles of rotation are taken as negative. The sense of the axes are such that the angles of rotation are positive when the directions of rotation are from -f-X to +Y, -f-Y to -{-Z, and -fZ to -f X. The axial and rotational conventions permit a particular cut of crystal to have the same rotation symbol for both right and left quartz. Refotiea Symbols 1-88. To specify the orientation of a piezoid cut, the following system, as recommended by the I. R. E. in 1949 is in general use. The crystal blank to be described is assumed to have a hypothetical initial position, with one corner at the origin of the coordinate system, and the thickness, length, and width lying in the directions of the rectangular axes. There are six possible initial positions, each of which is specified by two letters, the first letter indicating the thickness axis, and the second letter indicating the length axis. These positions are thus designated xy, xz, yx, yz, zx, and zy. The xy and yx positions are shown in figures 1-15 and 1-16, re- spectively. The starting position is so chosen that the final orientation may be reached with a mini- mum number of rotations. These rotations are taken successively about axes that parallel the z figun 1-15 xy Initial poshlon tor designating oriontation of crystal cwf WADC TR 55-156 19 SncHon I Physical Charadnristics of Pinsanlnclflc Crystals dimensions of the crystal at the time of rotation. Only the first rotational axis will coincide with a rectangular axis; however, the positive direction of any axis of rotation is that defined by the XYZ system for the initial position. A single rotation is sufficient for describing the majority of standard cuts, and three rotations is the maximum in any case. The dimensions and axes of rotation are indi- cated by the symbols, t, 1, and w, for thickness, length, and width, respectively. The Greek letters <f>, 0, and ^ designate the first, second, and third angles of. rotation, respectively. The following ex- ample, illustrated in figure 1-17, is a complete geo- metrical specification of a crystal plate : yztwl 30V15'’/25° t = 0.80 ± 0.01 mm 1 = 40.0 ± 0.1 mm w = 9.00 ± 0.03 mm The lettered combination at the beginning of the specification is called the “rotation symbol.” The first two letters, yz, of the symbol indicate the initial position, and the next three letters, twl, state the axes of rotation and the order in which the rotations are taken. The three angles, all posi- tive in this case, give the orientation and are listed in the same order as the respective rotations. The dimensions listed are those of the particular plate, and are not to be considered as necessary specifi- cations for that type of cut. For circular plates, the initial position will indicate which directions are to be considered thickness and length, so that the same rotation ssmibol is used as for rectangular z Figure 1-16 yx Initial pogitlon for designating oriontation of crystal tut Section I Standard Quartz Elements Figure 1-17 Orientation of crystal having the rotational specIHcations ystwl: 30°/15°/25'’ plates: in specifying the dimensions, however, 1 and w are replaced by the diameter. PIEZOELECTRIC ELEMENTS 1-89. The performance characteristics of a crystal plate are dependent on both the particular cut and the mode of vibration. For convenience, each "cut- mode” combination is considered a separate “pie- zoelectric element,” and the more commonly used elements have been assigned a letter sjnnbol. For example, the thickness-shear mode of the AT cut is designated as element A. STANDARD QUARTZ ELEMENTS 1-90. The principal quartz elements are given be- symbols listed first, low, with those which have been assigned element Element Symbol Name of Cut Rotation Symbol and Orientation Mode of Vibration Frequency Range in KC A AT yxl 35“21' or yzw 35'’21' thickness-shear 600 to 125,000 B BT or YT* yxl — 49®8' or yzw — 49®8' thickness-shear 1,000 to 76,000 C CT yxl 37°40' or yzw 37°40' face-shear 300 to 1,100 D DT yxl — 52‘’30' or yzw — 52*30' face-shear ' 60 to 500 E -!-6“X xyt 5* length-extensional 60 to 600 * The YT cut, which is essentially the same as the BT cut, was developed independently by Yoda in Japan. WADC TR 56-156 20 Section I Standard Quartz Eiomonts Element Symbol Name of Cut Rotation Symbol and Orientation Mode of Vibration Frequency Range in KC F -18.5°X xyt -18.5° length-extensional 60 to 300 G GT yxlt -51°7.5745° width-extensional 100 to 550 H 5°X yxt 5° length-width flexure 10 to 50 J Duplex 5°X xyt 5° (right quartz) and xyt 5° (left quartz) length-thickness flexure 0.4 to 10 M MT xytl 0° to 8.5°/±34° to ±50° length-extensional 50 to 500 N NT xytl 0° to 8.5°/±38° to ±70° length-width flexure 4 to 100 — AC yxl 31° or yzw 31° thickness-shear 1,000 to 15,000 — BC yxl —60° or yzw —60° thickness-shear 1,000 to 20,000 — ET yxl 66°30' or yzw 66°30' combination flexure and face-shear 600 to 1,800 — FT yxl —57° or yzw —57° combination flexure and face-shear 150 to 1,500 — V xzlw or xywl 15° to 29°/— 14° to -54° and 13° to 29°/27° to 42° thickness-shear 1,000 to 20,000 (fundamental) — V xzlw or xywl 0° to 30°/±46° to ±70° face-shear 60 to 1,000 — X xy length-extensional 40 to 350 — X xz width-extensional 125 to 400 — X xy or xz thickness-extensional 350 to 20,000 — Y yx or yz thickness-shear 500 to 20,000 TYPES OF CUTS 1-91. The standard quartz elements can be divided into two groups : in the first group belong those crystals which are most conveniently described as being rotated X-cut crystals, and in the second group belong those crystals which are most con- veniently described as being rotated Y-cut crystals. The first will hereafter be designated as the X group, and the second as the Y group. 1-92. The X and Y cuts have their thickness di- mensions parallel to the X and Y axes, respectively, with the length and width dimensions parallel to the two remaining axes. See figure 1-18. Thus, in describing a crystal orientation, the X cut is the equivalent of the two initial positions xy and xz, and the Y cut is represented by the initial positions yx and yz. Belonging to the X and Y groups, then, are those crystals whose rotation symbols begin with the letters x and y, respectively. As a general rule, from the X group, the low-frequency crystal units are obtained, and from the Y group, the medium- and high-frequency units. A third group of crystals is theoretically possible, where the initial position is a Z cut (thickness parallel to the Z axis) : however, because the piezoelectric effect Z AXIS Plgur 0 1-78 Orientation of X, Y, and Z cut plates WADC TR 56-156 21 Section I Standard Quartz Element* is restricted to the X and Y axes, the electrodes must be placed across one of these axes, which for the Z cut, would be at the edges — not a convenient location. Nor have other cuts, more or less simply oriented relative to a Z cut, been found to have optimum performance characteristics. However, there are experimental Z cuts, such as some of the ring-shaped crystals, which have proven of high quality, even though not practical for general use. The X Group 1-93. The principal crystals of the X group are listed below with the frequency ranges for which they have found commercial application : Name of Cuts Frequency Range in KC X 40 to 20,000 5°X 0.9 to 500 -18°X 60 to 350 MT 50 to 100 NT 4 to 50 V 60 to 20,000 z (B) 5*XCUT Figure 1-19 shows the orientations of an xy initial position (X cut with the length parallel to the Y axis) for the various cu^'-. X (O-ie'x CUT z' z X (0)MT CUT Ptgun 1-19. The X group. (Tho second rotations ot the MT, NT, and V cuts are shown only for the positive angies) * WADC TR 56-156 22 Section I Standard Quartz Element* THE X CUT 1-94. The X cut was the original quartz plate in- vestigated by Curie, and thus is sometimes called the “Curie cut.” This cut was also the first to be used as a transducer of ultrasonic waves and as the control element of radio-frequency oscillators. However, because of its comparatively large co- efficient of temperature, the X-cut plate is now rarely used in radio oscillators. As a transducer of electrical to mechanical vibrations, especially at high frequencies of narrow bandwidth, the X cut has a high electromechanical coupling efficiency, and is still widely used to produce ultrasonic waves in gases, liquids, and solids. These applications are largely for testing purposes, such as the measure- ment of physical constants and the detection of flaws in metal castings. 1-95. CHARACTERISTICS OF X-CUT PLATES IN THICKNESS-EXTEN SIGNAL MODE Description of Element: X cut; xy or xz; thick- ness-extensional mode. Frequency Range: 350 — 20,000 kc (fundamental vibration) ; lower frequencies when coupled as transducer for generating vibrations in liquids and solids. nk Frequency Equation: f = — ^ (n = 1, 3, 5, . . .) Frequency Constant: k., = 2870 kc-mm. Temperature Coefficient: 20 to 25 parts per million per degree centigrade; negative (i.e. for each degree increase or decrease in temperature, the frequency respectively decreases or in- creases 20 to 25 cycles for each megacycle of the initial frequency — a rise in temperature of 10°G would thus cause the frequency of a 5000-kc crystal to drop 1000 to 1250 cycles per second.) *Methods of Mounting: Sandwich and undamped air-gap — for oscillator circuit; transducer mounting depends upon particular type of mechanical load. Advantages: Mechanical stability, economy of cut, efficiency of conversion of electrical to me- chanical energy, and large frequency constant make this piezoelectric element preferred for the radiation of high-frequency acoustic waves when the ratio of the highest to the lowest frequency need not exceed 1.1. Disadvantages: Large temperature coefficient, tendency to jump from one mode to another, and the difficulty of clamping crystal in a fixed position without greatly damping the * See paragraphs 1-132 to 1-171. WADC TR 56-156 RATIO OF OPTICAL TO MECHANICAL AXIS Figure 1-20. Frequency constant for length-exten- sional mode (curve A) of X-eut crystal where the width and length are parallel to the Z and Y axes, respectively. Curve B is the frequency constant of a face-shear mode coupled to a second flexural mode, whose Interference makes the crystal useless for w/l ratios between 0.2 and 0.3, unless the thickness approaches the dimensions of the width * normal vibration prevent this element from being preferred for oscillator control. An electromechanical coupling factor of 0.096, which is only one-fourth that of the best syn- thetic crystals, makes this element inefficient as a radiator of a wide band of frequencies. 1-96. CHARACTERISTICS OF X-CUT PLATES IN LENGTH-EXTENSION AL MODE Description of Element; X-cut; xy; length-exten- sional mode. Frequency Range: 40—360 kc. 21 Ic Frequency Equntion: f = -y-’ (n = 1, 2, 3, . . .) Frequency Constant: Varies with w/l ratio— see figure 1-20. Temperature Coefficient: Negative**, varies with w/l ratio — see figure 1-21 ; zero coefficient if w/I = 0.272 and w = t. Methods of Mounting: Sandwich, air gap, wire, knife-edge clamp, pressure pins, cantilever clamp; more than one pair of electrodes re- quired for overtones ; transducer mounting de- pends upon particular type of mechanical load. ** All quartz bars have negative temperature coeffieienta for pure length-extensional vibrations, although a zero coefficient is obtainable for certain cuts. 24 Section i Standard Quartz , Clemenis W/Jl Figure 1-21. Temperature coefficient for length-ex- tensional mode of X-euf crystal, where w is parallel to the Z axis, and t = 0.051 Advantages: For w/1 ratios from 0.35 to 1.0, the fundamental length-extensional vibration is not strongly coupled to other modes, and hence the resonance is easily excited and of good stability except for drift during temperature variations. Although not preferred over zero- temperature-coefficient cuts, this element, with temperature control, is reliable for use in low-frequency oscillators, and for long, thin bars, for use in filters. However, its most important application is to produce ultrasonic vibration in gases, liquids, and solids, when the ratio of highest to lowest frequency need not exceed 1.1. Disadvantages: Inefficient as transducer of any but narrow frequency band, since electro- mechanical coupling is only one-fourth that of the better synthetic crystals. Strong coupling with a flexural mode makes the crystal use- less at w/l ratios between 0.2 and 0.3 (see figure 1-20) , and a weak coupling with a shear mode causes the frequency constant to de- crease as the w/l ratio approaches 1.0. This coupling to other modes interferes with the frequency response of the element when used in filters, unless the w/l ratio is 0.1 or less. Although for long thin bars the temperature coefficient is only about 2 parts per million per degree, this is greater than the minimum ob- tainable with fi^X-cut bars. 1-97. CHARACTERISTICS OF X-CUT PLATES IN WIDTH-EXTENSION AL MODE Description of Element: X-cut; xz; width-ex- tensional mode. Frequency Range: 126 to 400 kc. nlCi Frequency Equation: f = — ^ (n = 1, 2, 3, . . .) Frequency Constant: Varies with w/l ratio; see WADC TR 56-156 Figure 1-22. Frequency characteristics of X-cut crystal vibrating in width-extensional mode, where the width is parallel to the Y axis, w/l ratios not included between the two outer curves will have interfering modes. K, is in kc-inches figure 1-22, which shows the face dimensions that will have a single frequency near the de- sired resonance. Plates with dimensions not included between the two outer curves will have interfering modes. Temperature Coefficient: Negative ; approximately 10 parts per million per degree centigrade, but varies with w/l ratio. Methods of Mounting: Sandwich, air gap, wire, knife-edge clamp, pressure pins, cantilever clamp. Advantages: If cut with dimensions within the single-frequency range shown in figure 1-22, this element can be used in temperature-con- trolled low-frequency oscillators and narrow- band-pass filters. With the thickness dimen- sion ground for a particular high frequency, the same crystal unit may be used to generate either of two widely separate frequencies. Disadvantages: Relatively large temperature co- efficient prevents this element from being pre- ferred over the low-coefficient cuts. 25 Section I Standard Quartz Elements THE 5° X CUTS 284 1-98. The 5°X cut is the orientation that provides a zero temperature coefficient for the lengthwise vibrations of long, thin X bars, as shown in figure 1-23. Thus, this cut is preferred over the non- rotated X cut for use in low-frequency filters and control devices. Its length-extensional, length- width-flexural, and duplex length-thickness-flex- ural modes are defined as the elements E, H, and J, respectively ; the last named element, J, provid- ing the lowest frequencies. However, the 6°X ele- ments are also coupled to the other modes, so that for w/1 ratios much greater than 0.1 the frequency spectrum is little improved over that of the length- extensional mode of the X cut. Furthermore, as the w/1 ratio increases, so also does the temperature coefficient. For these reasons the 5°X elements are especially advantageous only when the w/1 ratio is 0.1 or less. These long, thin bars are used com- mercially for the control of low-frequency oscilla- tors and as filters, and are particularly adaptable for use in telephone carrier systems. 1-99. CHARACTERISTICS OF ELEMENT E Description of Element: 5°X cut; xyt: 6° ; length- extensional mode. Frequency Range: 50 to 500 kc. nlc Frequency Equation: f — (n = 1, 2, 3, . . .) Frequency Constant: Varies with w/1 ratio (see figure 1-24 )’. Temperature Coefficient: Varies with w/1 ratio (see figure 1-25, which holds for temperatures between 45 and 55 degrees centigrade). The -40 -30 -20 -10 0 10 20 30 40 90 ANGLE OF ROTATION AROUND THE X AXIS Figure 1-23. Temperature eoefUeieM for length-ex- tensional mode of long, thin X-group bars versus angle of rotation * Figure 1-24. FrequeiKy constant versus w/1 ratio for element E * frequency deviation of representative E ele- ments of different w/1 ratios is shown in fig- ure 1-26, where the initial frequency is taken at 26°C. Note that the temperature coefficient in parts per hundred per degree is the slope of a curve, and varies from positive to zero to negative as the temperature increases. Methods of Mounting: Wire, knife-edge clamp, pressure pins, cantilever clamp; more than one pair of electrodes required for overtones. Advantages: The low temperature coefficient and a large ratio of stored mechanical to electrical energy make this element 4 )ref erred for filter networks. Long, thin bars have only a very O 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 RATIO OF WIDTH TO LENGTH Figure 1-25. Temperature coefhclent versus w/1 ratio for element E at temperatures between 45° and 55° C* WAOC TR 56-156 26 Saction. I Standard Qvartx Elamantt fjgwra 1-28. Tomparature coefflWant varsws w/l rath for a/amant H * Figaro 1~26. Poreonlego froquottcy dovlathn for B olomonti of vartovi w/l ratios. Initial tmmporaturo — 25°C weak coupling to other modes and are used for both filter networks and low-frequency oscillators. If a w/l ratio greater than 0.15 is desired, a ratio of approximately 0.39 is opti- mum insofar as a low temperature coefficient is concerned. Disadvantages: At v//l ratios between 0.2 and 0.3 the length-extensional mode is so closely coupled to the length-width flexure that the crystal is useless; as the width is increased the coupling of the length-extensional to the face-shear mode becomes stronger, and the temperature coefficient becomes larger. How- ever, because of the large electro-mechanical coupling of this element, w/l ratios of 0.35 to 0.5 can still be favorably used in filters if a temperature coefficient less than 4 parts per million is not required. 1-100. CHARACTERISTICS OF ELEMENT H Description of Element: 5°X cut; xyt; 5° ; length- width flexure mode. Frequency Range: 10 to 100 kc. Figaro 1-27. Froqaoney constant vorsus w/t ratio for olomont H Frequency Equation: f = nksW/1’, (n = 1, 2, 3, ...) Frequency Constant: Varies with w/1 ratio (see figure 1-27). Temperature Coefficient: Varies with w/l ratio (see figure 1-28). Methods of Mounting: Wire, in vacuum; free-free' flexures of long, thin bars have nodal points for the fundamental vibration at a distance of 0.224 X 1 from the ends; two electrically opposite pairs of electrodes are plated on each side of the YZ faces, with “ears” at the nodal points for soldering to the mounting wires. See figure 1-29. When the polarity of the lower electrodes causes a contraction of the bar, the polarity of the upper electrodes causes an ex- tension, and vice versa — the over-all result being a flexural deformation. Advantages: For long, thin bars the length-width flexural mode is resonant at much lower fre- quencies than is the length-extensional mode. This advantage, combined with the favorable electro-mechanical coupling, and reasonably low temperature coefficient, has made this ele- ment useful in very-low-frequency filters where only a single frequency is to be selected. When mounted in vacuum, a Q of 30,000 is obtainable. Figaro 1-29. Blomont H, showing division of oloctrodo plating for oxciting furtdamontal rnodo. Similarly divided oloctrodos are on reverse side. Tho nodal "oars" whom tho mounting wins are attachod, am at a distanco of approximatoly 0.224 times tho longth from tho ends * WAOC TR 56-156 27 FREQUENCY CONSTANT IN KILOCYCLE-CENTIMETERS Section I Standard Quartz Elements Figure 1-30. Frequency constant versus t/l ratio for element J RATIO m/i Figure ?-3l. Percentage frequency deviation for J elements. The smaller t/l ratio is representative of a 1.2-kc element, and the larger t/l ratio Is repre~ sentative of a lO-kc element Figure 1-32. Frequency constant versus w/l ratio for various resonances of — 18° X-cut crystal. A Is the width-extensional mode. B is the face-shear mode, which, at small w/l ratios, is strongly coupled to D, the second flexural mode. C represents the band between the antiresonant (upper curve) and the res- onant (lower curve) frequencies of the length-exten- sional mode of element F. Note the weak coupling between C and D-B * WADC TR 56-156 28 Section I Stondard Quortx Eiementa Disadvantages: The ratio of stored mechanical to electrical energy is not as large as that of the length-extensional mode, and because of this, the element does not give as broad a band-pass spectrum. Also, the elfect of the shear stresses causes the temperature coefficient to become highly negative as the w/1 ratio is increased. Finally, the damping effect of the air is greater for flexural than for other vibrations, so that flexure crystals should be mounted only in evacuated containers. 1-101. CHARACTERISTICS OF ELEMENT J Description of Element: Duplex 5°X cut; xyt: 5° (right quartz), and xyt: 5° (left quartz); length-thickness flexure mode. Frequency Range: 0.4 to 10 kc. Frequency Equation: f = nk,t/l* (n = 1, 2, 3, .. .) Frequency Constant: Varies with t/1 ratio (see figure 1-30). Temperature Coefficient: Varies with both the t/1 ratio and the temperature ; figure 1-31 shows the total relative frequency deviation of two elements of different t/1 ratios, the initial fre- quencies being those at 26°C. The temperature coefficients in parts per hundred at a given temperature are the slopes of the curves at that point. Note that the temperature at which a zero coefficient is obtained increases as the t/1 ratio decreases. At temperatures below that of a zero-coefficient point, the coefficient is positive ; at temperatures above, it is nega- tive. Methods of Mounting: Headed-wire, in vacuum; two thin plates are cemented together with polarities opposed so that only one pair of electrodes, plated on opposite YZ faces, are required ; the crystal element is supported at the nodal points, which for the fundamental vibration are at a distance 0.224 x 1 from each end. Advantages: Small temperature coefficient and low resonant frequencies (among the lowest obtainable with quartz) make this element useful in providing stable control for sonic- frequency oscillators, and as a component of single-frequency filters. Disadvantages: Not economical for control of fre- quencies above 10 kc. 1-102. CHARACTERISTICS OF ELEMENT F Description of Element: — 18.5°X cut; xyt: —18.5° ; length-extensional mode. Frequency Range: 60 to 300 kc. tik Frequency Equation: f = -j-^ (n = 1, 2, 3, . . .) Frequency Constant: Varies slightly with w/1 ratio (see figure 1-32). Temperature Coefficient: 25 parts per million per degree centigrade — varies very little with changes in the w/1 ratio. Methods of Mounting: Wire, knife-edge clamp, pressure pins, cantilever clamp ; more than one pair of electrodes required for overtones. Advantages: The extremely weak coupling of this element to the face-shear and second flexure modes, represented by curves B and D, re- spectively, in figure 1-32, permits a better frequency spectrum than can be obtained with element E for w/1 ratios greater than 0.1. For this reason, the F element used to be pre- ferred over the E element as a filter plate, and was the principal quartz element in the channel filters of coaxial telephone systems. This is no longer true because channel filters now use -j-5°X plates which are smaller and conserve quartz. Disadvantages: Relatively large temperature co- efficient prevents this element from being pre- ferred for oscillator control or as a channel filter if wide variations in temperature are to be expected. Also, the F plate is larger than the E plate of the same frequency and thus consumes more quartz. 1-103. CHARACTERISTICS OF ELEMENT M Description of Element: MT cut; xytl : 0° to 8.5°/ ±34° to ±50°; length-extensional mode. Frequency Range: 50 to 500 kc. Tik Frequency Equation: i — — p (n = 1, 2, 3, . . .) Frequency Constant: Varies with w/1 ratio and angles of rotation (see figure 1-33) . Temperature Coefficient: Varies with w/1 ratio and angles of rotation (see figure 1-34), and with the temperature. The total relative fre- quency deviation of an 8.5°/±34° M element, where the initial frequency is taken at 40°C, is shown in figure 1-35. Note that the tem- perature coefficient, which is the slope of the curve, changes from positive to negative as the temperature increases, with the zero co- efficient occurring at 63°C. Methods of Mounting: Wire, knife-edge clamp, pressure pins, cantilever clamp; more than one pair of electrodes required for overtones. Advantages: The MT crystals were developed in an effort to overcome the large negative tem- perature coefficients of the X-cut and the 5°X- WADC TR 56-156 29 Section I Standard Quarts Elomontt cut length-extensional modes for the larger w/1 ratios. See figures 1-21 and 1-26. The un- favorable temperature characteristics are caused by the coupling of the extensional to the face-shear mode, the latter having a high negative temperature coefficient. However, if the crystal is rotated about its length, an orientation will be found where the face-shear mode has a zero temperature coefficient ; that is, the coefficient will pass from negative to positive values. The low temperature coeffi- cient of the length dimension will thus be pre- served even though the coupling to the shear- mode has not, itself, been diminished. The low temperature coefficient makes the M ele- ment advantageous for oscillator control in the 50-to-100 kc range, and for use in narrow band filters, such as pilot-channel filters in carrier systems, where wide temperature ranges are to be encountered. The 8.5'’/34® rotation with a w/1 ratio of approximately 0.42 provides the greatest electromechanical coupling of the M elements, and hence the broadcast bandpass of the MT cut for use in filters. Disadvantages: The electromechanical coupling rapidly decreases as the w/1 ratio increases, so that at ratios greater than 0.7 the element is too selective for filter use, and of too small a piezoelectric activity to be advantageous for oscillator control. Maximum electromechani- cal coupling is obtained with w/1 ratios of 0.39 to 0.42 ; but for a maximum bandwidth the E element is preferred. Although the in- terference of the face-shear temperature co- efficient is reduced, the coupling to that mode remains relatively strong ; so where the tem- perature varies very little, or where the sec- ondary frequency effects are undesirable, the F element is preferred. 1-104. CHARACTERISTICS OF ELEMENT N Description of Element: NT cut; xytl: 0° to 8.5”/ ±38° to ±70”; length-width flexure mode. Frequency Range: 4 to 100 kc. Frequency Equation: f (n = 1, 2, 3, . . .) Frequency Constant: Varies with w/1 ratio (see figure 1-36). Temperature Coefficient: For w/1 ratios of 0.2 to 0.5, low coefficients are obtained by double rotations of 0” to -|-8.5”/±50”. Typical fre- quency deviation curves are shown in figure 1-37, where the initial temperature is taken at 25°C. Note that a zero temperature coeffi- RATtO OF WIDTH TO ceNOTH Figure 1-33. Frequency constant versus w/1 rath for M olomonts having low temperature coeFRelents. C is the curve of the most commonly used MT orientation * UJ Figure 1-34. Temperature coefficient versus w/1 ratio for M elements * dent occurs at approximately 10”C. To pro- duce a zero temperature coefficient at 25°C for w/1 ratios of 0.05, the angles of rotation should be as shown in figure 1<^8. Methods of Mounting: Wire, in vacuum; special characteristics are the same as for the H ele- ment. See paragraph 1-100. WADC TR 56-156 30 S«cMon I Standard Quarts ElamanI* TEMPERATURE IN DEGREES CENTIGRADE figure 1-35, Freqwency-tomperatwre eharac- t»ri$ties of ahmant M * TEMPERATURE CO Figura 1-37. Fraquancy-tamparatura eharactarlstiea of alamant N. Tha largar w/l ratio Is typical of 100-kc alamanis, and tha smallar w/l ratio la typical of 16-kc alamants Figura 1-36. Fraquancy constant varsus w/l ratio for alamant N Figura 1-38. Anglos of rotation for N ala- mant with a w/l ratio of 0.05 which will provida xaro tamparatura coafficiaat at 35°C * Advantages: The principal advantage of the N element is that the second rotation reduces the temperature coefficient for the flexure vi- bration of long, thin crystals. This is accom- plished by changing the width from near parallelism to the Z axis to near parallelism to the X axis. Theoretically the ideal rotation would be 90°, except that the piezoelectric effect would be reduced to zero. As a compro- mise, secondary rotations, about the length, of 39° to 70° are made. Besides reducing the flexure-mode temperature coefficient of the long, thin crystals, the rotation also reduces the negative coefficient for the shear modes at the higher w/l ratios, as in the case of the M element. Where wide temperature ranges must be met, this element is preferred for very-low frequency oscillators, and ‘in single- frequency filter selectors. As the control ele- ment of an oscillator, it can maintain the frequency within ±0.0026% over a normal room-temperature range without temperature control. Disadvantages: The electromechanical coupling is rather weak, more so for the larger than for the smaller w/l ratio. As a consequence, the bandwidth is too narrow for the element to be used as a band-pass filter of communica- tion channels, and the piezoelectric activity is so low that special circuits are required for its use in oscillators. WAOC TR 56-156 31 Section I Standard Quartz Elomantz THE V CUT 1-105. The V cut, developed by S. A. Bokovoy and C. F. Baldwin of RCA, is actually an entire series of cuts obtained by a sequence of double rotations of an initial X-cut plate. The first rotation angle, 0 , is taken about the Z azis, and the second rota- tion angle, 6, is taken about the Y' axis (the di- mension of the crystal that is initially parallel to the Y axis). For each angle there is an angle 9 at which the crystal will have a given temperature coefficient for a particular mode of vibration. Nor- mally, the combination of angles desired is one that will provide a zero temperature coefficient; however, it may be that a small positive or nega- tive coefficient is required to counterbalance an opposite temperature coefficient inherent in the external circuit to which the crystal is to be con- nected. For this purpose curves of $ plotted against <t>. are shown in figures 1-40 to 1-41 for small posi- tive and negative temperature coefficients, as well as for a zero temperature coefficient. Other ^ and 6 combinations may be extrapolated to give tem- perature coefficients differing from the actual values shown. It should be noted that when the rotation about the Z axis is equal to ±30°, the thickness dimension becomes parallel to a Y axis, and hence the crystal is in the position of the Y cut, with the Y' axis coinciding with an X axis. Thus, if <^ = ±30°, the V cut is essentially the same as a rotated Y cut, and in this case would embrace practically the entire Y family. On the other hand, if 4 . = 0°, the V cut becomes simply a singly rotated X cut — but with rotations about the Y axis, not the X axis as in the case of the 5°X and the — 18°X cuts. However, when <f> = 0°, the V cut does overlap the MT and NT cuts. 1-106. CHARACTERISTICS OF V-CUT PLATES IN THICKNESS-SHEAR MODE Description of Flement: V cut; xslw or X 3 rwl: 15° to 29°/-14° to -54° and 13° to 29°/27° to 42° (see temperature coefficient curves in fig- ure 1-40 for exact <i> and 9 combinations) ; thickness-shear mode. Frequency Range: 1000 to 20,000 kc (funda- mental) ; higher frequencies on overtones. Frequency Equation: f = (fundamental vibra- tion when t << 1 and w). Figure 1-39 shows the frequency constant of the zero-tempera- ture-coefficient series of V cuts as a function of the first rotation angle. The upper curve, designated k, ( -f e) , applies to positive angles of 6, the second rotation, whereas the lower curve, designated k, (—6), applies to nega- tive angles of 9. Temperature Coefficient: Figure 1-40 shows the combinations of ^ with positive values of 9 that provide temperature coefficients of -|-15, 0, and —15 parts per million per degree centi- grade, and those combinations of ^ with nega- tive values of 9 that provide temperature coefficients of -f 5, 0, and —5 parts per mil- lion per degree centigrade. Methods of Mounting: Sandwich, air gap, clamped air-gap, button. Advantages: The principal advantage of the V cut is that a given temperature coefficient may be obtained from a large choice of orienta- tions, and with a minimum in trial-and-error procedure. Not only can a series of zero- coefficient plates be obtained, but also plates with coefficients of desired sign and magni- tude for annulling the known frequency-tem- Ftgun 1-39. Frmgueney constant versus (angle of rotation about Z axis) for the thickness-shear mode of V-eut crystals when 9, the second angle of rotation, is so chosen that a zero temperature coefficient Is obtained. The upper and lower curves are for positive and negative values of 9, respectively WADC TR 56-156 32 Saction I Standcmi Quarts EUmant* perature effects of the circuits in which the plates are to be used. The V cut is the only member of the X group that provides a zero temperature coefficient for high-frequency vibrations ; and because of the large choice of rotation angles, one or the other of the V orientations will frequently permit the maxi- mum use of an unfaced or badly twinned mother crystal. Because their larger fre- quency constants permit a thicker and less fragile crystal, the orientations with a nega- tive 6 are preferred for the higher frequen- cies. Also, small deviations in negative values of 6 produce less variation in the temperature coefficient than do the same deviations in posi- tive values of 6 . Hence, the negative orienta- tions of 6 are also generally more dependable for obtaining a desired temperature coeffi- cient. On the other hand, positive values of 6 permit a less bulky crystal for the lower fre- quencies, a less critical frequency constant, less interference from spurious frequencies, and for accurately determined orientations, a broader temperature deviation for a given deviation in frequency. At <^ = 30°, the values of = —49°, -t-31°, and -|-35°3r are substan- flgur* 1-40. kmtatlona of 0 to for tMcknoss-sfcoar mod* of V eat, which prwido tho tomporaturo coofficionts imfieotod for ooch curve tially the same as the BT, AC, and AT cuts, respectively, of the Y group, as described in paragraphs 1-114, 1-111, and 1-112. The. chief use of the thickness-mode V cut is for the control of high-frequency oscillators. Disadvantages: The possibility of spurious fre- quencies close to the desired fundamental is the most troublesome limitation of the V cut operating in a thickness-shear mode. As a general rule, the coupling between the desired and the stray modes diminishes as the initial rotation <t> is increased. At values of less than 13°, the interference is too great for sta- ble operation. Because of the relatively poor frequency spectrum, the V cut is not readily adaptable for use in selective networks. With a certain amount of cut-and-try experimenta- tion, the more objectionable modes may be reduced by grinding down the width and length dimensions. For angles of 4 , close to 30° the length and width dimensions most important to avoid are approximately the same as those given in paragraphs 1-112 and 1-114 for the AT and BT cuts, respectively. 1-107. CHARACTERISTICS OF V-CUT PLATES IN FACE-SHEAR MODE Description of Element: V cut; xzlw or xywl: 0° to 30°/±45° to ±70° (see temperature co- efficient curves in figure 1-41 for exact 4 and 6 combinations) ; face-shear mode. Frequency Range: 60 to 1000 kc. Frequency Equation: f = k,/w (fundamental for square plates). Frequency Constant: Insufficient data exist to plot the curve of k« for all the combinations of and 6 corresponding to this element. However, in the case of the zero-coefficient plates, as the positive value of 6 approaches 37.5°, k, approaches 3070 kc-mm, and as the negative value of 9 approaches —62.5°, ki approaches 2070 kc-mm. Temperature Coefficient: Figure 1-41 shows the combinations of 4> and 9 that provide tem- perature coefficients of -f 6, 0, and —5 parts per million per degree centigrade. Methods of Mounting: Wire, cantilever clamp. Advantages: The principal advantage is the low temperature coefficient, which makes the ele- ment useful for low-frequency oscillators and filters. The large choice of orientation angles is also advantageous for obtaining the maxi- mum number of cuts from a given mother crystal, particularly if the presence of twin- ning or other defects limit the dimensions in WAOC TR 56-156 33 section I Standard Quartx ElomanH Figure 1-41. Relations of $ to if>, for fate-shear mode of V cut, which provide temperature coeRIcients of 0, -|-5, and — 5 parts, per million per degree centigrade the directions at which rough bars would normally be cut. Also, the angles for small predetermined positive and negative coeffi- cients permit a crystal to be cut which can exactly annul the known temperature effects of the external circuit. As indicated in figure 1-41, small deviations in the orientations angles will cause minimum deviations in the temperature coefficient when 0 = 0° to 15°, and $ is negative. On the other hand, maxi- mum piezoelectric activity is obtained when <t> is large, and 0 is positive. As a general rule, the positive values of 6 are used for the higher' frequencies and the negative values of 6 for the lower frequencies. The zero-temperature cuts for <t> = 30° are substantially the same as the CT and DT cuts of the Y gproup. See paragraphs 1-116 and 1-116, respectively. Disadvantages: Care must be taken that flexure modes are not strongly coupled to the face- shear mode. Such coupling may be reduced by making the plates square, or nearly so. For angles of approaching 30°, the thickness should be approximately within the limits given for the C and D elements in paragraphs 1-115 and 1-116. Tka Y Grovp 1-108. The principal crystals of the Y group are listed below with the frequency range for which they have found commercial application : Hame of Cut Frequency Range in KC Y 1000 to 20,000 AC 1000 to 15,000 AT 500 to 100,000 BC 1000 to 20,000 BT 1000 to 75,000 CT 300 to 1100 DT 60 to 600 ET 600 to 1800 FT 150 to 1500 GT 100 to 660 Figure 1-42 shows the orientations of a yx initial position (Y cut with the length parallel to the X axis) for the various cuts. In special cases the width may be parallel to the X axis, but this is the exception rather than the rule, unless the plate is square or circular. With the exception of the GT cut, the crystals of the Y group are used in their shear modes — face shear for the low-frequency elements, and thickness shear for the high-fre- quency elements. The Y cut, itself, has a large positive temperature coefficient; and, because of coupling between the thickness-shear mode and the overtones of the face-shear mode, it also ex- hibits sharp irregularities in its frequency spec- trum. However, by rotation about the X axis, zero temperature coefficients may be obtained, and the coupling between the shear modes can be greatly diminished. This coupling becomes zero at the angles of the AC and BC cuts, and the frequency constant of the thickness-shear mode has minimum and maximum values, respectively, for these two orientations. Figure 1-43 shows the thickness- shear frequency constant, and figure 1-44 the thickness-shear temperature coefficient, with each plotted as a function of the angle of rotation. For the face-shear mode, the frequency constant and the temperature coefficient are shown in figures 1-45 and 1-46, respectively, plotted as functions of the angles of rotation. THE Y CUT 1-109. The Y cut was introduced commercially in the late 1920’s, at which time its principal advan- tage was that it could be clamped at its edges, whereas the X cut would not oscillate if the edge movement were even slightly restricted. The use of a Y cut, vibrating in a shear-mode, was origi- nally suggested by E. D. Tillyer of the American Optical Company, to whom a U. S. patent was WADC TR 56-156 34 SMtion I Stondard Quaiix Elvmvnt* Figure 1-42. Rotation anglos of Y cut about X axis which provide the principal members of the Y group. The GT cut is the only member having a second rotation (±45° about the Y' axis). The *'S" indicates that the positive end of the X axis points toward the observer issued in 1933. For this reason, the Y cut is some- times called the Tillyer cut. For several years this crystal was used extensively in commercial and military transmitters mounted in mobile equip- ment, and also in commercial broadcast transmit- ters where the Y cut's readily excited oscillations permitted the use of crystal oscillators with low plate voltages. However, due to the strong cou- pling between the thickness-shear and the over- tones of the face-shear and flexure modes, the Y cut’s frequency spectrum is very poor. Also, small irregularities in the dimensions of the crystal readily produce abrupt changes in the frequency. A typical frequency-temperature curve of a Y-cut crystal is shown in figure 1-47. Today, the Y cut has been almost entirely replaced by the rotated cuts having small temperature coefficients, and the Y cut’s only major application now is that of transducer for generating shear vibrations in solids. 1-110. CHARACTERISTICS OF Y-CUT PLATES IN THICKNESS-SHEAR MODE Description of Element: Y cut; yx or yz; thick- ness-shear mode. Fre(/uency Range: 500 to 20,000 kc; much lower WADC TR 56-156 35 Sectipn I Standard Quartz Elamants ROTATION ABOUT X AXIS - -Wi f »•+ ROTATION ABOUT X AXIS IN DEGREES IN DEGREES Figure 1-43. Frequency constant versus angle of rotation about X axis for thickness-shear elements of Y group. (Values shown for Y, BT, and BC cuts are smaller than the average) * so -TS -60 -4S -30 -IS 0 IS 30 4S 60 7S BO ANGLE OF ROTATION ABOUT X AXIS IN DEGREES ( t ) Figure 1-44. Temperature coefficient versus angle of rotation about X axis for thickness-shear elements of Y group * WADC TR 56-156 36 Section I Standard Quarts Eloinonts > K it 3E»0 •-8 iu 300 0 ^ u >2 ZTSO 8:i ,.i tsoo z ' Su 1210. 15 - a n ■ K m Gi IS m ■ 3 Si Wo 2000 c o tk .J IS 22 m ■ n ►0 -7S -so -AS -30 -IS 0 IS 30 45 SO TS »0 “ ORIENTATION ANCLE IN DEGREES 1^) Figurm 1-45. FnqtMHcy constant vorsut attgh of rota- tion about X axis for faco-$hoar olomontt of Y group frequencies when bonded to solids for use as transducer. Frequency Equation: f (fundamental vibra- tion) . Frequency Constant: k, = 1981 kc-mm (average value) . Temperature CoeffUient: Varies with dimensions of crystal and with temperature but is usually between 76 and 125 parts per million per de- gree centigrade, and is positive, with an aver- age value of 86 parts per million per degree centigrade. Methods of Mounting: Sandwich ; air gap, clamped air gap ; bonded to solids when used as trans- ducer. Advantages: Ratio of stored mechanical to elec- trical energy is larger than that of any other TEMPERATURE Figure 1-47. Tomporaturo-froquohey charactorislics typical of tho Y-eut,, thkknoss-shoar olomont. Yho froquoney jumps aro most apt to occur whan small discropauclos aro prossat In tbo thkknoss-cllmonsioH Figure 1-46, Tomporaturo coofHclont versus angle of rotation about X axis tor face-shear elements of the Y group quartz element; this large ratio, combined with the quartz crystal’s superior strength, makes the Y cut desirable as a generator of shear vibrations in solids for the purpose of measuring or testing the solids’ physical properties. This element is the easiest of all quartz cuts to excite into vibration, and thus requires the lowest voltages for operation. Large temperature coefficient makes element useful as a sensitive detector of variations in temperature. Disadvantages: Large temperature coefficient, dis- continuities of resonant frequencies, and poor frequency spectrum make this element a sec- ondary choice for use in either oscillator or filter circuits. Special Y cuts, such as the block- and doughnut-shaped crystals in figure 1-48, vibrate in a combination mode com- posed of coupled shear and flexure modes, and have zero temperature coefficients at cer- tain temperatures. However, because of the prevalence of spurious frequencies, the large volume of quartz used per cut, and the diffi- culties of mounting, these crystals have little practical use. Figure 1-48. Y-cut block and doughnut-shaped crys- tals which can provide zero temperature coefBclents for certain combination modes WADC TR 56-156 37 Section I Standard Quartz Elomontt 1-111. CHARACTERISTICS OF AC-CUT PLATES IN THICKNESS-SHEAR MODE Description of Element; AC cut; yxl: 31° ; length- thickness-shear mode. Frequency Range: 1000 to 15,000 kc (fundamental vibration) . Frequency Equation: f = (fundamental vibra- V tion when t < < 1 and w) . Frequency Constant: k, = 1656 kc-mm. Temperature Coefficient; 20 parts per million per degree centigrade; positive. Methods of Mounting: Sandwich, air gap, clamped air gap, button. Advantages: This element vibrates in a very pure length-thickness mode with an excellent fre- quency spectrum. It has the lowest frequency constant of all the quartz thickness modes and thus permits a smaller thickness, and hence a more economical cut, for use at the low end of the high-frequency spectrum. For a given temperature, the electrical parame- ters of an AC crystal unit can be predeter- mined with an accuracy equal to, or greater than, that of the more commonly used AT units. Disadvantages: The principal disadvantages of the AC cut is its relatively large temperature coefficient; because of this the element has found little commercial use, and the low-co- efficient AT cut, with, an orientation close enough to that of the AC for the coupling be- tween the shear modes to be small, is generally preferred. 1-112. CHARACTERISTICS OF ELEMENT A Description of Element: AT cut; yxl: 36°21'; length-thickness-shear modes; or, }rzw: 36° 21'; width-thickness-shear mode. Frequency Range: 500 to 1000 kc (special cuts) ; 1000 to 15,000 kc (fundamental vibration) ; 10,000 to 100,000 kc (overtone modes). Frequency Equation: f = ks/t (fundamental vibration when t< <1 and w) f + ajf (P- 1) w* 2 where m, n, and p are integers. Frequency Constant: k, = 1660 kc-mm. Temperature Coefficient: 0.0 at 26°C; figure 1-49 shows the total relative frequency deviation for the normal maximum, minimum, and aver- age angles of this element; the temperature coefficient at each point on a curve is the slope at that point in parts per hundred. At ^ = 36°16', the temperature coefficient will vanish at 45°C, changing from negative to positive as the temperature increases. Opti- mum orientations for zero coefficients at other temperatures are given below : Deg. C ♦ 20 35°18' 20 35°27' (overtones) 75 35°31' 76 35°33' (overtones) 85 85°83' 86 36°86' (overtones) 100 36’36' 190 (max.) S6°26' Methods of Mounting: Sandwich, air gap, clamped air gap, button. Advantages: The excellent temperature-frequency characteristics make this element preferred for high-frequency oscillator control wher- ever wide variations of temperature are to be encountered; it is particularly applicable -60 40 to 0 . to 40 60 60 *100 TCMPCRATURE (*0) Figure 1-49. Temperafure-froquency tharactorlstles of o/omont A 38 WADC TR 56-156 for aircraft radio equipment where sharp changes in tonperature may be frequent, but where the added weight of constant'tempera- ture ovens is undesirable. The angle of orien- tation is sufficiently close to that of the AC cut for the coupling between the shear modes to be weak, so that the resonant frequency can be isolated from that of other modes, ex- cept for certain dimensions of w and 1. The A element also shares the low frequency con- stant of the AC element, and this is preferred for frequencies at the low end of the high- frequency spectrum. However, because of its superior temperature-frequency and piezo- electric characteristics as compared with the BT characteristics, the A element may well be preferred for the control of frequencies in the vhf range, even though the BT cut has the larger frequency constant. Disadvantages: Because of the large thickness di- mensions that would be required, the A ele- ment is generally not economical for the gen- eration of frequencies below 1000 kc, although special circular cuts have been used at fre- quencies as low as 500 kc. In its normal high- frequency range, the most troublesome prob- lem is to find the proper length and width dimensions which will permit the desired fre- quency to be widely separated from other modes. Although the orientation of the AT cut is close to that of the AC cut, there still exists a fair amount of coupling to the face- shear modes, and to the extensional and flex- ural modes along the X axis. Sufficient infor- mation is not available to avoid a certain amount of trial and error in grinding and fin- ishing an AT blank to provide an optimum frequency spectrum at a desired frequency; however, there are certain X and Z' (the di- mension parallel to the Z axis before rotation) values that can be avoided by use of the fre- quency equations which hold approximately for the less complex of the unwanted modes. The following equations give the face dimen- sions of an AT cut which will produce un- wanted resonances at the same frequency, f, as the thickness-shear mode. For extensional modes along X: X = 2488 n f (n = 1, 3, 5, . . .) For flexure modes along X : X=: 1338.4 n f (n = 2, 4, 6, Section I Standard Quartz Elements For shear modes along X ; V 2542.0 n (n = l, 8, 5,...) For shear modes along Z': Z'= 2540.0 n (nrri, 3, 5, ...) With f expressed in kc, X and Z' are given in mm. Either X or Z' may be the length, with the other dimension being the width. It has been found that the unwanted modes are somewhat restricted by giving the plate a con- vex contour, and also by the use of circular plates. The convex contour is possible for all but the very thin plates that are used at fre- quencies above 15,000 kc. A 1000-kc crystal may have a contour of 3 to 5 microns. The equations above hold for flat plates, and be- come increasingly in error as the contour is increased. 1-113. CHARACTERISTICS OF BC-CUT PLATES IN THICKNESS-SHEAR MODE Description of Element: BC cut; yxl: —60°; length-thickness-shear mode. Frequency Range: 1000 to 20,000 kc (fundamental vibration). k Frequency Equation: ^ ~ (fundamental vibra- tion when t < < 1 and w) . Frequency Constant: kj = 2611 kc-mm. Temperature Coefficient: 20 parts per million per degree centigrade; negative. Methods of Mounting: Sandwich, air gap, clamped air gap, button. Advantages: The advantages and disadvantages of the BC cut are similar to those of the AC cut, except that the BC thickness-shear fre- quency constant is the highest obtainable for a rotated Y cut. A BC cut may thus have a greater thickness for the same frequency, and hence be less likely to be shattered from over- drive or mechanical shock — a distinct advan- tage at the higher fundamental frequencies where very thin crystals are used. Since the BC orientation is the negative angle of rota- tion which provides zero coupling between the shear modes, the element vibrates in a very pure length-thickness mode with an excellent frequency spectrum. For a given temperature, the electrical parameters of a BC crystal unit can be predetermined with an accuracy equal to or greater than that of the more commonly used BT units. Disadvantages: As in the AC cut, the principal disadvantage of a BC cut is its relatively large WADC TR 56-156 39 SMtion I Standard Quartz Elamants temperature coefficient. Because of this, the element is not widely used, and the zero-co- efficient BT cut, with an orientation suffi- ciently near to that of the BC cut to have a weak coupling between the shear modes, is used instead. An added disadvantage is that the magnitude of the rotation away from the Y axis is approximately double that for the AC cuts. For this reason the piezoelectric co- efficient is smaller for the BC than for the AC or AT cuts, and, hence, somewhat higher voltages are required to maintain oscillations. 1-114. CHARACTERISTICS OF ELEMENT B Description of Element: BT cut; yxl: — 49°8'; length-thickness-shear mode ; or, yzw : — 49°8' ; length-width-shear mode. Frequency Range: 1000 to 20,000 kc (fundamental vibration) . 16,000 to 75,000 kc (overtone modes). Frequency Equation: f ks/t (fundamental vibration when t « 1 and w) f = h* n‘ + ai -p- -1- aj (P - D* w* where m, n, and p are integers. Frequency Constant: k, = 2560 kc-mm. Temperature Coefficient: 0.0 at 25°C; figure 1-50 shows the total relative frequency deviation for the normal maximum, minimum, and average angles of this element; the temperar ture coefficient in parts per hundred per de- gree centigrade at each point on a curve is the slope at that point. Zero coefficients are obtained at 20°C and 76°C when ^ is — 49®16' and — 47°22', respectively. Methods of Mounting: Sandwich, air gap, clamped air gap, button. Advantages: The temperature-frequency charac- teristics make this element useful for high- frequency oscillator control where the tem- perature is not expected to vary too widely from the mean value. It is particularly ap- plicable for use in radio equipment which is to operate at the high end of the high-fre- quency spectrum. Most of the high-frequency crystal oscillators employ either the BT or the AT cut, with the B element, because of its larger frequency constant, often preferred at frequencies from 10 to 20,000 kc. Since the orientation angle is near that of the BC cut, the shear modes are not too strongly coupled together; and, when ground to proper dimen- sions, the B element exhibits a reasonably satisfactory frequency spectrum. Disadvantages: Like the A element, the B element is not suitable for use at the lower frequencies because of the large thickness dimensions - «0 40 ZO C to 40 *0 ao 4 100 TEMPCKATURE (*C) WADC TR 56-156 figure 1-50. Tumporuture-truguencY eharecturMcs of olemonf B 40 that would be requii'ed. Because of its grreater angle of rotation from the Y axis, the BT has a smaller piezoelectric coefficient than the AT cut, and hence requires a higher voltage to maintain oscillations. Also, the temperature coefficient of the BT cut increases more rapidly than that of the AT cut when the temperature varies to either side of the zero point. Moreover, zero temperature coefficients cannot be obtained at as widely separated temperature, as can be done with the AT cut by slightly varying the orientation angle. This limitation, however, becomes an advantage in- asmuch as it reduces the percentage error when cutting a crystal to provide a given tem- perature-frequency characteristic. The great- est problem in preparing a B element is to avoid those length and width dimensions that cause the frequencies of unwanted modes to approach the frequency of the desired mode. As in the case of the AT cut, a BT blank with a good frequency spectrum will require a cer- tain amount of trial and error in the finishing process. For the simpler modes of lower order, the following equations give the face dimen- sions of a BT cut which produce unwanted resonances of the same frequency, f, as that of the thickness-shear mode. For flexure modes along X: „ 1810 n , n A c \ X — ^ (n — 2, 4, 6, • . .) For shear modes along X : For shear modes along Z'l Z'= (n = 1, 3, 5, . . .) Section i Standard Quartz Elements With f expressed in kc, X and Z' are given in millimeters. {Z' is the dimension of the ro- tated Y cut that originally was parallel to the Z axis.) Either X or Z' may be the length, with the other dimension being the width. As in the case of the A element, a convex con- tour of a plate will aid in restricting unwanted modes. At 1000 kc the contour may be as great as 5 microns ; the thin, 20,000-kc plates, however, must be flat. The equations above hold only for flat plates, but are approxi- mately correct if the contour is very small. 1-115. CHARACTERISTICS OF ELEMENT C Description of Element: CT cut; yxl or yzw: 37® to 38° ; face-shear mode. Frequency Range: 300 to 1100 kc. Frequency Equation: f = ki/w (fundamental of square plate). Frequency Constant: k, = 3070 kc-mm. (Square plates are preferred since they have fewer secondary frequencies.) Temperature Coefficient: 0.0 at 25° C for rotation angle of 37°40'. Figfure 1-51 shows the total relative frequency deviation with tempera- ture for maximum, minimum, and average angles of rotation for a nominal cut of 37°40'; the initial temperature is taken at 25°C. The slope of a curve at any point is the tempera- ture coefficient in parts per hundred per de- gree centigrade at that point. Note that as the rotation angle is increased, the zero co- efficient is shifted to a higher temperature; TEMPERATURE (*0 WAOC TR 56-156 figure 1-51. Temperature-frequmHcy characteristics of element C 41 Section I Standard Quartz Elements Z' the same is true when the ^ ratio is in- creased. For square plates, zero coefficients can be obtained at higher temperatures (50°C to 200°C) by rotation angles from 38‘’20' to 41° 50', respectively. Methods of Mounting: Wire, cantilever clamp. Advantages: The CT cut is essentially a BT cut rotated approximately 90° so that the face shear of the C element corresponds to the thickness shear of the B element. This orien- tation provides a zero-temperature-coefficient shear mode for generating low frequencies, without requiring a crystal of large thickness dimension. The frequency characteristics of the C element, as compared with the D ele- ment, are roughly analogous to those of the B with the A element, except that the former pair vibrate at low frequencies, and the latter at high frequencies. The C element has the higher frequency constant, so is generally preferred over the D element at the high end of the low-frequency range. The C element is widely used both for low-frequency oscillator control and in filters, and does not require constant temperatures control under normal operating conditions. One of its principal ap- plications has been as the control element in frequency-modulated oscillators. Disadvantages: Because of its larger frequency constant, the C element must be cut with larger face dimensions than the D element to provide the same frequency of vibration. Thus, for the generation of very low frequen- cies the smaller DT cut is the more economical to use. Care must be taken that flexure modes are not strongly coupled to the face-shear mode. To prevent a coincidence of resonance between the two modes, the following thick- nesses have been used : Frequency Range in KC 870 to 428 428 to 476 475 to 540 730 to 876 876 to 1040 Thickness in Mils 18.6 to 19.9 16.0 to 17.6 18.6 to 19.9 12.0 to 14.0 16.0 to 17.6 1-116. CHARACTERISTICS OF ELEMENT D Description of Element: DT cut; yxl or yzw: —62° to —53° ; face-shear mode. Frequency Range: 60 to 600 kc. Frequency Equation: f = )^/'n (fundamental of square plate). f + W* (m (n 1. 2, 3, 1 , 2 , 8 , ) ) Frequency Constant: ki = 2070 kc-mm. (Square plates are preferred since they, have fewer secondary frequencies.) Temperature Coefficient: 0.0 at 26°C for rotation angle of — 52°30'. Figure 1-62 shows the total relative frequency deviation with tempera- ture for maximum, minimum, and average angles of rotation for a nominal cut of — 62°30', where the initial temperature is taken at 26°C. The slope of a curve at any point is the temperature coefficient in parts per hundred per degree centigrade at that point. Note that as the rotation angle is in- creased, the zero coefficient is shifted to a higher temperature. The upper limit for a zero coefficient is approximately 200°C, when ^ = -64°. Methods of Mounting: Wire, cantilever clamp. -«0 40 20 0 20 40 eo ao 4-IOO TEUPCSATURe CO WADC TR 56-156 Figure 1-52. Tumpurature-fruquuiKy charaetuHsties of olomont 0 42 Section I Standard Quartz Elamontt Advantages: The DT cut is essentially an AT cut rotated approximately 90° so that the face shear of the D element corresponds to the thickness shear of the A element. This orien- tation provides a zero-temperature-coefficient shear mode for sreneratinsr low frequencies, without requiring a crystal of large thickness dimension. Because the frequency constant is less than that of element C, the face dimen- sions of element D are smaller for a given frequency, and hence the DT is the more eco- nomical cut for use at very low frequencies. Like the C element, the D element is widely used in both oscillators and filters, and does not require constant temperature control under normal operating conditions. Disadvantages: At frequencies above 500 kc, the impedance effects introduced by the mounting become excessive, since the contact surfaces between the crystal and the supporting wires become rather large compared with the area of the crystal faces. Hence, the C is preferred over the D element in the 500 — 1000 kc range, since the higher frequency constant of the former permits a larger crystal face. To avoid strong coupling of the face-shear mode with flexure modes, certain thicknesses must be avoided. For most frequencies, however, a thickness of approximately 17 mils is satis- factory. 1-117. CHARACTERISTICS OF ET-CUT PLATES IN COMBINATION MODE w Description of Element: ET cut, with -y ratio ap- proximately equal to 1.0; yxl or yzw; 66°30'; combination of coupled modes with second flexural vibration appearing to dominate a face-shear harmonic. Frequency Range: 600 to 1800 kc. Frequency Equation: f = k/w (square plate). 2k f = T-T- — (nearly square plate). I yv Frequency Constant: k = 5360 kc-mm. Temperature Coefficient: 0.0 at 75° C; see figure 1-53 for total relative frequency deviation. Methods of Mounting: Wire; preferably mounted in vacuum. Advantages: Besides its zero temperature coeffi- cient, the principal advantage of the ET cut is its high frequen <7 constant, which is almost 1.8 times that of the C element. This permits an effective extension of the frequency range for this type of plate and mounting. Optimum performance is obtained at temperatures near 40 SO 60 70 SO 90 K>0 110 TEMPERATURE IN DEGREES CENTI6RADE Figure 1-53. Temperature-fregueney tharaeterlsties of ET-cot plate vibrating in combination mode * 75°C, so that the element is particularly ad- vantageous where crystal ovens are used. Disadvantages: Stability and general performance are inferior to those that can usually be ob- tained by using, according to the particular frequency, either an A or a C element. 1-118. CHARACTERISTICS OF FT-CVT PLATES IN COMBINATION MODE Description of Element: FT cut, with w/1 ratio, approximately equal to 1.0; yxl or yzw; —57°: combination of coupled modes with second flexural vibration appearing to domi- nate a face-shear harmonic. Frequency Range: 150 to 1500 kc. Id Frequency Equation: f = — (square plate) ; w 2k f = ; (nearly square plate). 1 + w Frequency Constant: k = 4710 kc-mm. Temperature Coefficient: 0.0 at 76°C; see figure 1-64 for total relative frequency deviation. Methods of Mounting: Wire ; preferably mounted in vacuum. Advantages: The advantages of the FT cut are approximately the same as that of the ET, except that the FT has a lower frequency constant. The FT is related to the ET in ap- proximately the same way that the DT is WADC TR 56-156 43 Section I * Standard Quartz Elements 40 SO 60 70 eo 90 100 110 TEMPERATURE IN DEGREES CENTIGRADE Figure 1-54. Temperature~trequency charaeterfstics of FT-cut plate vibrating In cembinrrflen mode * related to the CT. However, the frequency constants of the £T and FT are approxi- mately twice that of the low-frequency shear elements, so that these cuts can be made in practical sizes for twice the frequencies ob- tainable from the CT and DT crystals. Like the ET, the FT cut is particularly suited for use in ovens at temperatures between 70® and 80“C. temperature in degrees c Figure 1-55. Temperature-frequency characteri§tlc$ of element C '* Disadvantages: Stability and general performance are inferior to those that can usually be ob- tained by using, according to the particular frequency, either an A, C, or a D element. 1-119. CHARACTERISTICS OF ELEMENT G w Description of Element: GT cut, with ratio -y = 0.859; }7xlt: 61°7.6'/46°; width-extensional mode. Frequency Range: 100 to 660 kc. Frequency Equation: k,/w (fundamental). Frequency Constant: ki = 8870 kc-mm. Temperature Coefficient: Very nearly zero over the range from —26® to 4-76®. Figure 1-66 shows the total relative frequency deviation from the initial frequency at 0®C. Note that ■1 ■ ■ ■■ ■1 Kill ■1 ■ ■ ■ ■■ !■ II ■r ■ ■ ■ ■ ■ ■■ ■■ ■■ 1 ■ mi 11 m ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■■ ■■ ■■ ■■ ■■ ■ 1 ■ rll ■1 ■ ■ ■J ■ ■ ■ ■ ■ ■ ■■ ■■ ■■ ■■ II II ■ -90 -60 -30 0 SO 60 90 FIRST ROTATION ANGLE, IN DEGREES (B) Figure 1-55. (A) Diagram Illustrating the equivalence betvreen a face-shear mode and the length- and wldth-extenslonal modes of a rectangular plate which has been cut diagonally with respect to the face-shear element. (B) w/l ratio vs rotation angle, 5 , of element G providing sere temperature coefficient * WADC TR 56-156 for a span of lOO^C the frequency does not vary more than one part in a million from the center frequency. The midpoint of the flat portion of the curve can be shifted from 25°C to SO’C by increasing the initial orientation angle from 61®7.6' to 61‘’30'; the zero coeflS- cient range will thus extend from 0°C to 100°C. A temperature variation of ±15°C on either side of the midpoint will not change the frequency moreihan 0.1 part in a million. Methods of Mountinp; Wire, knife-edge clamp, pressure pins, cantilever clamp. Advantages: The GT cut provides tiie greatest frequency stability that has yet been obtained from a quartz plate. Where other quartz ele- ments have zero temperature coefficients at only one temperature, the G element will not vary more than one part in a million over a range of lOO’C. This element was originally suggested by the fact that a face-shear mode consists of two extensional modes coupled to- together. When a face-shear element is rotated 45” the vibrations lose their shear effect and appear as two extensional modes — one along the width, and the other along the length. See figure 1-56 (A). Since all pure extensional modes must have a negative or zero tempera- ture coeflSdent, a positive coefiicient of a face- shear mode must be due to the coupling be- tween its two extensional components. If the cut of a face-shear crystal having a positive coefficient has been rotated 45”, the coupling between the extensional modes can be reduced by grinding down one edge so that one of the modes will increase in frequency. As the fre- quencies become more widely separated, the extensional modes will approach their true FABRICATION OF 1-120. The production stages during the fabrica- tion of a crystal unit may differ somewhat from ' one manufacturer to the next because of variations in the instruments, techniques, and the type of mountihg employed. However, the general pro- cedure is fundamentally the same throughout the industry — ^first, the inspection and cutting of the raw quartz; next the lapping and etching of the diced blanks; and finally, the mounting and testing of the crystal unit in its holder. INITIAL INSnCTION OF RAW 9UARTZ 1-121. The manufacture of a crystal unit begins WADC TR 56-1S« Sedien I Fabrication of Crystal Units negative temperature coefficients. At some ratio of width to length a zero coefficient will be obtained. The GT cut is properly a d:46” rotation of any positive-coefficient face-shear crystal in the Y group. Although the most satisfactory cut is the one described above, a number of other GT cuts have been investi- gated where the initial angle of rotation, has had negative as well as positive values. F.igure 1-66 (B) shows the w/1 ratios for both positive and negative angles that will provide a zero temperature coefficient. For negative values of the dominant mode is the one of lower frequency, whereas for positive angles of ^ the higher-frequency mode is dominant. The G element is used for the control of oscil- lators where the most precise accuracy is re- quired, such as in the frequency standards of loran navigational systems, the time stand- ards at the U. S. Bureau of Standards and at Greenwich Observatory, and in both fixed and portable frequency standards of general use. Other than in frequency and time standards, the GT cut is employed in filters that are de- signed for use under very exacting phase conditions. Disadvantages: The principal disadvantage of a GT cut is its expense. To obtain optimum temperature - frequency characteristics re- quires pains-taking labor in cutting and grinding to the exact orientation and dimen- sions. Furthermore, the excellence of a par- ticular cut will be of little advantage unless the mounting and the external circuit are also of superior design. For these reasons a G ele- ment is not particularly practical except when the utmost frequency precision is mandatory. CRYSTAL UNITS with the inspection of the raw quartz for impuri- ties, cracks, and inclusions. For this purpose, the arc lamp of the inspectoscope shown in figure 1-57 is used. 1-122. The inspectoscope tank is filled with a clear, colorless oil mixture having an index of refraction approximately the same as the average in quartz (1.52 to 1.56). Such a medium for transmitting the light to and from the raw crystal, or “stone,” is necessary in order to see the interior, for otherwise reflections and refractions at the rough surface will not only create an excessive glare, but will diminish the intensity of light penetrating beyond. 4S ..:pY|TAi^Nij fabrication Section I Fabrication of Crystal Units iw.A»Dfcsc?N.t' V eWTCH rtiCAftoEseewT, tAltP SWITCH U -".t; I f'S'^Vo.iiC /' j.t.tTivjtf*:', Figun 1-57. Po/ar/scope-/nspocfoscopo. Used tor oxaminlng raw quartz The lamp incorporates a high-powered (500- to 1000- watt) projection system of white light, and inspection of the stone is performed by direct ob- servation. The usable parts of the stone are marked ; or if too many imperfections are present, the stone is discarded. INSPECTION FOR OPTIC AXIS AND OPTICAL TWINNING 1-123. If a stone has retained some of its natural faces, the optic (Z) axis may be readily located by direct observation. In the usual case, however, it is necessary to use the plane-polarized light system that is provided by the inspectoscope. The light from a mercury or incandescent lamp is plane polarized by a Polaroid plate placed between the lamp and the tank. On the opposite side of the tank is a second Polaroid plate with its transmission axis perpendicular to that of the first, so that if a stone is not in the tank to rotate the polarity of the light, the rays will be stopped at the second plate. Light that does filter through, however, is re- flected upward by a mirror, and the pattern may be observed through the glass cover shown in figure 1-57. When a stone is placed in the tank and oriented so that its optic axis is parallel to the rays, the polarity of the rays will be rotated and a bright WADC TR 56-156 image will be reflected from the mirror. If white light is used, a pattern of concentric rainbow colors will appear ; and if monochromatic light is used, a pattern of concentric rings of light and darkness will appear. The optic axis will be exactly parallel with the light rays when the stone is in the posi- tion that yields the fewest and broadest bands. If optical twinning is present, it will be revealed by a fine-toothed pattern cutting across the rings, as indicated in figure 1-58. The twinning areas are more clearly indicated when white light is used, and when viewed slightly off the optic axis. On the other hand, monochromatic light produces ring patterns of maximum clarity for the determina- tion of the optic axis itself. Flat surfaces are ground on opposite sides of the stone, parallel to the optic axis ; and, with the stone resting on one of the flat surfaces at the bottom of the inspecto- scope tank, a line is drawn on the upper surface to indicate the approximate Z-axis direction. USE OF CONOSCOPE FOR EXACT DETERMINATION OF OPTIC AXIS 1-124. After the approximate optic (Z) axis is determined, the stone is cemented to a glass plate, and a small-end-section of the crystal is sliced off with a diamond saw, leaving a flat surface approxi- 46 /// nc ' Section I Fabrication of Crystal Units mately perpendicular to the optic axis. The stone is then mounted on an adjustable orienting jig, which is placed against the reference edge in a conoscope tank. The conoscope (see figure 1-59) provides a polarized light system with which the optic axis may be accurately located by observing a concentric ring system. The principle of the conoscope is similar to that of the polarizing sys- tem of the inspectoscope, except that a converging lens system and a vernier system are provided that permit the optic axis to be determined with an accuracy of one degree. The handedness of the crystal is readily determined by rotating the second Polaroid plate, or analyzer, of the conoscope. The quartz is right or left according to whether the concentric rings appear to expand or contract for a given direction of analyzer rotation. When the Z axis is accurately determined, each end is trimmed to form plane surfaces (“windows”) exactly perpendicular to the Z direction. SECTIONING THE STONE 1-126. There are three general methods of cutting the stone to obtain crystal blanks of desired orientation: the direct-wafering, X-block, and Z- section-Y-bar methods. In direct watering, shown in figure 1-60, wafers are sliced directly from the stone at the desired orientation, and the blanks are diced from the wafer. The X-block method, as in- dicated in figure 1-61, is similar to that of direct Figure t-58. Polarized-light view of pyramidai cap indicating optical twinning * figure T-59. Conoscope. Used for locating accurately the optic axis end for determining the handedness of quartz stones * WADC TR 56-156 47 , III stc. Sadion I PabricQtien of Cryctal Unita wafering, except that, before being sliced into oriented wafers, the stone is cut into one or more blocks with place surfaces at each end of the Z axis', and at the ends of one of the X axes; each surface is accurately cut at right angles to the axis it terminates. It is from these “X” blocks that the properly oriented wafers are cut and then diced into blanks. The third method of cutting proceeds as indicated in figure 1-62. The stone is sliced into Z sections (cross-sectional slabs with plane faces perpendicular to the Z axis) ; the Z sections are cut into Y bars (bars with the length parallel to the Y axis) ; and crystal blanks are sliced at the desired Figure 1-40. Dintt-wafaring method of cutting ciyataf Monk* MOTHER OUARTZ SECTIONED INTO Z BLOCKS RLANE OF PAPER) PLACED AGAINST REFERENCE EDGE OF MOUNTING JIG. WAFER CUT FROM X BLOCK FIgum 1-41. X-block method of cutting wafers from unfoced stone. Mfofers, on being diced, provide cryctal blankt at the proper orientation WAOC TR 56-156 49 Section I Fabrication of Cryctal Unit* 2 SECTION LHQ MATCHING WINDOW LHQ MATCHING ARM RHQ MATCHING ARM RHQ MATCHING WINDOW marking template MARKING ARM QUARTZ SP^IMEN FILAMENT LAMP HOUSING NOTE. LHQ»LEFT-HfcND 0O*NTZ, RHO« RIGHT-HAND QUARTZ Figun 1-^3. Pinhole orlaseopo with mofehinfl ana marKing arms tor use on Z sections WADC TR 56-156 Section I Fabrication of Crystal Units M Figure 1-64. X-ray deferminafion of X axis in Z block. M is horizontal bisector of angle that ray must make if reflected beam is to enter ionization chamber. 6 , the Bragg angle of X-ray reflection for copper- anode Ka radiation, is predetermined according to the particular atomic plane to be identified. For plane that is parallel to an m face, and hence to an X axis, $ — 10° 38'. With positions of X-ray source and ionization chamber fixed, rotation of Z block about Z axis will cause maximum current to flow through ionization chamber when an X axis becomes perpendicular to M Figure 1-65. Reflection patterns of twinned Z section, showing both types of twinning. The section is pre- dominantly right quartz, but is fairly evenly divided by the electrical twins a and b. The small regions of optical twinning of one electrical sense are shown in C, and those of the opposite sense are shown in D. The X-axis polarities indicated apply only to the respective bright regions. The regions marked f contain flaws * figure 1-64, an X-ray beam is directed toward the crystal’s vertical surface, which deflects part of the beam into the window of an ionization chamber, causing a current to flow that has an amplitude proportional to the intensity of the rays entering the chamber. X-rays of constant wavelength are propagated in a narrow pencil from a properly filtered source, which consists of a special high- voltage cathode-ray tube having a copper anode. The X-rays are emitted by virtue of the high- energy electrons’ striking the copper target, and a thin nickel plate is inserted in the X-ray path to eliminate unwanted wavelengths. The atomic planes of the crystal lattice effectively serve as reflecting surfaces, except that interference be- tween the reflected rays from adjacent parallel planes eliminates all angles of reflection except those that permit the path lengths of coinciding rays to differ by an integral number of wave- lengths. The above condition is satisfied when the distance between the atomic planes is related to the wavelength and the angle of incidence of the X rays in a manner that can be expressed by Bragg’s law : nA = 2d sin 6 where: n = 1, 2, 3, A = wavelength of X rays d = distance between parallel atomic planes e = angle of incidence of X rays with atomic plane ’The ionization chamber is a gas-filled metallic cylinder having an electrode which is maintained at a voltage relative to the cylinder. X rays enter- ing the chamber will ionize the gas, permitting a current to flow through the external circuit. With 6 predetermined for a particular atomic plane, the exact direction of the plane, and hence of the crystal’s orientation, can thus be determined by rotating the Z block for a maximum reading on the ammeter. CUTTING X BLOCK 1-128. When the X direction has been precisely determined, the mounting jig is locked in position and transferred to a diamond saw, where windows are cut perpendicular to, and at each end of, the X axis — thereby forming an X block. After the align- ment of the windows is rechecked with the X-ray apparatus, the X block is cleaned, and then etched in 48% hydroflouric acid or a saturated solution of ammonium difluoride. WADC TR 56-156 50 DETERMINATION OF TWINNING 1-129. Electrical and optical twinning boundaries can be observed directly by shining a spot lamp upon the etched Z windows of the X block. The light should be directed at approximately a 30- degree angle with the surface being examined, with the line of sight of the observer perpendicular to the surface. As the block is rotated about the axis perpendicular to the surface, there will be four particular orientations, each corresponding to a reflection of maximum brightness from the etched area of one of the four possible twins — right-hand quartz of either electrical sense, and left-hand quartz of either electrical sense. See figure 1-65. Normally a crystal is predominantly right or left, so that optical twinning usually appears only in small scattered regions. Electrical twinning, how- ever, normally divides a crystal into large regions of opposite electrical sense. The polarities of the various twinned areas can be readily determined by noting the angles of rotation at which maxi- mum brightness is observed. The axial polarities of an X block may also be determined by examin- ing the X windows with the aid of a pin-hole oria- scope having matching and marking arms designed especially for X sections. The images observed will differ according to the electrical sense of the particular area — also, according to whether hydro- fluoric acid or ammonium difluoride was used in etching. By a proper interpretation of the patterns. Sactien I Fabrication of Crystal Units the axial directions of the twinned regions can be suitably marked. If there is an excessive amount of scattered twinning, the block must be dis- carded ; otherwise, the observation permits a proper orientation for cutting slabs, so that opti- mum use of the quartz is possible. PREPARATION OF WAFERS 1-130. The mounting jig, adjusted to the correct orientation, is transferred to a saw, and the X block is sliced into slabs of sufficient thickness for finishing. See figure 1-66. After being cleaned and etched, the slabs are inspected and marked for twinning, and the unusable portions are cut away by a diamond saw. Each slab is cemented to a holder and mounted in a jig for a final X-ray deter- mination of the orientation. The adjusted slab holders are transferred to the jig of a lapping machine, and the slabs are lapped on one surface, using an abrasive of 400-grain carborundum, until the lapped faces have the desired orientation. The slabs are then cemented to a large plate with the corrected faces down, and the uncorrected faces are lapped until parallel with the bottom faces. The “wafers,” as the slabs are now called, are next cemented to a glass-topped steel plate for dicing. PREPARATION OF CRYSTAL BLANKS 1-131. The wafers are diced to the approximate crystal blank size with a dicing saw, as shown in Figur* 1-66. Diamond saw for cutting wafers from X block WADC TR 56-156 51 Saction I Fabrication of Crystal Units CRYSTAL (WRK HOLDER (C) figure 1-67. (A) Dicing saw. (B) Diced wafer. (C) Nest of lapping machine. (D) lapping machine figure 1-67. The dice are then transferred to the nest of a lapping machine, where they are lapped to a thickness equivalent to several kilocycles be- low the desired frequency. The lapping proceeds in three stages: rough, semifinishing, and finishing. However, the rough stage is accomplished prior to dicing, if the slabs are first lapped to wafers as described in paragraph 1-130. The semifinishing is done with 600-grain carborundum or equivalent, and the finishing requires 1000- to 1200-grain ab- rasive. Each of the last two stages should com- pletely remove the surface left by the preceding stage. (Where extreme care is required, as when finishing thin harmonic mode plates, 3000 mesh aluminum oxide mixed with water, cosmetic talc and powdered white rouge provides high-precision results, with the ultimate operating dependability greatly increased if the final lapping is followed by figure 1-68. Loaf of crystal dice, all blanks oriented in the same direction in preparation for edge grittding by edging machine * a brief semi-polishing with a mixture of water, rouge, and small amounts of rust preventative and wetting agent.) In the case of high-frequency blanks, the final lapping should bring the blanks within 25 to 50 kilocycles of the desired frequency. Next, a stack of 25 to 100 dice are clamped into a loaf, with all the blanks oriented in the same direc- tion. See figure 1-68. Two exposed edges are then ground parallel to locating surfaces by an edging machine. The loaf is then reversed and the two remaining edges are ground to square the blanks. Finally, the blanks are etched to the proper fre- quency. For high-frequency crystals, a frequency tolerance of ±0.002 percent will require that the finished blanks be etched to within ±0.00001 milli- meter of the ideal thickness. After cleaning, the crystal blanks are ready for mounting. METHODS OF MOUNTING CRYSTAL BLANKS IN CRYSTAL HOLDERS 1-132. In the past, some confusion has resulted among radio engineers because of a mixed usage of the terms crystal holder and crystal unit by manufacturers in describing and naming their products. However, it is now generally agreed that the term crystal holder is to be used only in reference to the mounting and housing assembly, whereas the term crystal unit is to designate a complete assembly — that is, a crystal holder con- taining a mounted crystal plate. 1-133. Crystal holders have been variously classi- fied by different specialists in the field, and in the WADC TR 56-156 52 absence of a standard nomenclature, a certain amount of overlapping has resulted among the dif- ferent classifications. The procedure to be followed here is to treat each particular method of mount- ing as a separate category. The holders to be dis- cussed are described according to the following types of mounting: gravity-sandwich, pressure- sandwich, gravity-air-gap, comer-clamped-air-gap, nodal-clamped-air-gap, dielectric-sandwich, plated- dielectric-sandwich, button-sandwich, knife-edge- clamp, pressure-pin, cantilever-clamp, solder-cone- wire, headed-wire, and edge-clamped. Only two general classifications of mounting, wire and pres- sure, are specified for Military Standard crystal units currently recommended for use in equip- ments of new design. The wire-mounted crystals are cemented directly to supporting wires. The pressure-mounted crystals are clamped in place by frictional contact with electrodes or other sup- porting elements. The wire mounts include the solder-cone-wire, the headed-wire, and the ce- mented-lead types, the latter being a particular kind of edge-clamped support cemented to the crystal. The pressure mounts include all other types listed above except the gravity-sandwich and the gravity-air-gap. Gravity Sandwich 1-134. A “crystal sandwich” is simply a crystal plate sandwiched between two flat electrodes. In the simple gravity type of holder the crystal plate is placed on one electrode, with the second elec- trode resting on top and connecting to the external circuit through a flexible wire. See figure 1-69. A small clearance is provided around the sides to permit the crystal to vibrate freely. The clearance must not be too large, however, else the crystal will slide around in the holder, and may become chipped, or, at least, cause the electrical constants of the crystal unit to vary. The electrodes must be perfectly plane and made of noncorrosive metal, such as stainless steel, brass, or titanium. Brass is inferior to the other two metals, and titanium is largely a future possibility. The top electrode is considerably lighter than the bottom electrode, and is usually specifically dimensioned to fit a par- CRYSTAL^ • FLEXIBLE WIRE » 1 1 4- \_ NON CORROSIVE 1 METAL PLATES Flgun t-69. Gravity sandwkh SocHon I Fabrication of Crystal Units ticular crystal size. If the top electrode is too heavy, the impedance it introduces will be exces- sive, preventing the crystal from vibrating near its normal mode; on the other hand, if the top elec- trode is too light, firm contact will not be made with the crystal, and the operation of the crystal unit will be unstable. The edges of the crystal are slightly rounded to insure that they are free of burrs. Both the crystal and the electrodes must be perfectly clean, and the entire unit must be mounted in a hermetically sealed chamber. Nor- mally, the grid terminal of the unit connects to the flexible wire of the top electrode, and the ground or cathode terminal to the bottom electrode. 1-135. The grravity holder was at one time widely used, but has now been largely replaced by holders that can maintain the crystal in a relatively fixed position if subjected to external vibration, such as might be encountered in vehicular or aircraft equipment. Occasionally, even when mounted in vibration-free equipment, a gravity crystal unit may fail to operate because one edge of the crystal has become closely pressed against one of the sides of the chamber. However, a light tap of the holder is usually sufficient to start oscillations. Compared with the holders in which flat inflexible electrodes are pressed against the crystal, the activity of the gravity holder is generally superior, and requires less voltage to maintain a state of oscillation. Prastnra Sandwich 1-136. In holders of the pressure-sandwich type, the crystal is held more or less firmly between two flat electrodes under the pressure of a spring. In a typical assembly, the electrodes, which normally are of identical size and shape, are in turn sand- wiched between two very thin contact plates. The contact plates connect to two metal prongs that serve as electrical terminals and plugs for mount- ing the crystal holder in a socket. An insulating washer is placed over one of the contact plates, a coil spring is placed over the washer, and a neo- prene gasket is placed between the spring and the cover to provide hermetic sealing. Except for the glass spacers, the pressure holder described above is very similar to the air-gap holder shown in figure 1-70. 1-137. Although the activity of a pressure-type crystal sandwich normally is not as great as that of the gravity type, it is much preferred because of its greater ruggedness and less critical require- ments concerning the orientation and vibration of the equipment in which it is to be mounted. Another advantage of the pressure holder is its relative simplicity of design — fewer of its compo- WAOC TR 56-156 53 Section I Fabrication of Crystal Units 6LASS SPACER QUARTZ OSCILLATOR PLATE IIEOPRENE FLAT ELECTRODE (A) Figure 1-70. Methods, old and new, tor mounting crystal units. (A) Construction of early model crystal unit employing the gravity air-gap type of mounting, now largely outmoded. The crystal holder shown is the type FT-243. IBI Solder-cone wire mount for v-l-f len^h-ftexure crystal. (Courtesy HBEMCOI. (Cl Recently developed techniques for mounting shock-proof, 1-mc, A elements in the miniature HC-6/U metal holder to meet the specifications for 1-mc CR-IS/U crystal units: a. Reeves-Ho0man flexible nylon mount, b. Hupp loose-slotted edge-clamped mount, c. Bliley molded nylon bumper mount, d. RCA edge-clamping spring mount. (Courtesy McCoy Electronics.) WADC TR 56-156 54 nents require separate and exact dimensions for each particular frequency than is true of the ma- jority of holders. If the holder is constructed so that the sprinjr pressure may be adjusted, very slight variations in the frequency are possible ; the activity, however, will decrease proportionally as the pressure is increased. Although the pressure holder is widely used and is less expensive than most of the other types of holders, it has the dis- advantage of low activity and comparatively large damping of the oscillations. Thus, crystal units of this construction are not as sharply selective, nor as electrically predictable, as crystal units of more critical design. Gravity Air-Gap Mauating 1-188. A gravity air-gap crystal unit is essentially a gravity sandwich, hut witti an air space separat- ing the crystal from the upper electrode. The air gap may be variable or fixed. In the variable type, the frequency can be adjusted slightly by raising or lowering the upper electrode, by means of a screw. The fixed air gap, however, is more com- monly used. As shown in figure 1-70, the fixed gap is maintained by glass or other insulating spacers placed between the electrodes. It is important that the thickness of the air gap not be an even quarter- wavelength of the acoustic vibrations which the crystal will generate in the air. Otherwise, the air waves, on reflection from the upper electrode, will return to the crystal 180 degrees out of phase with the normal vibrations, thereby introducing a high impedance and greatly reducing the activity. Maxi- mum activity is obtained when the air gap is an odd quarter-wavelength in thickness. The exact dimension, however, is not particularly critical in the case of shear modes, and a variation ot ±^i wavelength will cause little change in the ampli- tude of the vibrations. The quarter-wavelength formula is -r = -n, where v is the sound velocity in air, and f is in cycles per second. At room tem- perature and pressure, v = 330,000 mm/sec = 12,992,000 mils/sec. The gap thickness should not exceed 3 mils, else the piezoelectric coupling will be too weak to maintain oscillations at reasonable voltages. Where it is necessary to have as large a piezoelectric coupling as possible, the air gap must be reduced to the minimum of ^ wavelength. 1-139. The advantage of the air-gap mounting is that it shares the simplicity of design of the sand- wich holders, but eliminates the damping effect caused by the frictional contact of the upper elec- trode with the crystal. The presence of the air gap also effectively inserts in the crystal circuit a SocHen I Fabrication of Crystal Units series capacitance equal to that between the upper electrode and the crystal face, thereby increasing the ratio of the stored electrical to the stored me- chanical energy, and thus decreasing the electro- mechanical coupling. The reduction of the fric- tional losses (i.e., the effective electrical resistance) and the electromechanical coupling serves to give the crystal unit a higher Q, and thus to make it more selective and stable, and less affected by small variations on the external circuit. However, with the decrease in electromechanical coupling, the tendency of the crystal to vibrate is reduced ; and also reduced is the bandwidth for use in filters. On the credit side, the gravity-air-gap mounting is particularly convenient for the preliminary test- ing of crystals at the time they are being ground, and it has also been widely used in fixed-plant equipment with thickness-mode crystals operating at frequencies between 200 and 1500 kc. The prin- cipal disadvantages of the gravity-air-gap holder are the loose mounting of the crystal, the reduced piezoelectric coupling, and the possibility that a momentary overdrive will cause arcing across the air gap, or a corona discharge, thereby damaging the crystal and electrode surfaces, or even punc- turing the crystal completely. Coraer-Clampod, Air-Gap Mountiag 1-140. The principal features of the corner- clamped, air-gap mounting are indicated in figure 1-71. Note that air gaps exist at both the major faces of the crystal, except at the corners where the electrode risers, or. lands, clamp the crystal in position. If the length of a thickness-shear plate, such as an AT or BT cut, is not less than twenty times the thickness, a firm pressure may be ap- plied at the corners without greatly reducing the activity. The same precaution for avoiding an air column with dimensions approaching a multiple of a half-.wavelength is necessary for the clamped as for the undamped holder. The lands at the cor- ners normally provide a gap of 0.5 to 2 mils. 1-141. The corner-clamped, air-gap method is widely used for mounting high-frequency thick- ness-shear crystals. Its operating characteristics Figure f-71. Typical ceriiar<lamp*d, air-gap mathod af maunling crystal* * WADC TR 56-156 55 S«c<ien I Fabrication of Cryatal Units are similar to those of the unclamped holder; and, in addition, it has the important advantage of clamping the crystal in a fixed position, thus per- mitting its use in aircraft and vehicular equip- ment. However, the clamping at the comers intro- duces an excessive amount of impedance when used for the lower-frequency, thickness-shear crystals where the 1/t ratio is less than 20 ; hence, this type of mounting is generally confined, to crystals with frequencies above 1500 kc. Nodal-Clamped, Air-Gap Mouatiag 1-142. The principal features of the nodal-clamped, air-gap mounting are indicated in figure 1-72. ibis method may be used for mounting low-frequency piezoelectric elements vibrating in an extensional mode and having a nodal area at the center of the crystal. Each electrode has two risers for clamp- ing the crystal at each end of its nodal axis, and thus provides a secure mounting with a minimum of damping from direct contact between the crys- tal and the electrodes. A general advantage of any “zone-type” clamping, such as the nodal or comer methods where particular areas of a crystal are subjected to pressure, is that spurious frequencies requiring free vibrations in the clamped zones will be suppressed. Dielectric Saadwich 1-143. This type of holder is essentially a crystal sandwich with a “lettuce” of high dielectric mate- rial inserted between the crystal and the elec- trodes. The sandwich and air-gap holders previ- ously described do not permit a crystal to operate FIgun 1-72. Typkal nodal-etampad, air-gap malhod of mouHfing ciystafs FIguro 1-73, Frossuro typo of dloloctrle sondw/cJi for mounting crystals near its elastic limit, for otherwise arcing would occur between the electrodes and the crystal. Low drive levels arc particularly necessary at frequen- cies above 4000 kc, for the likelihood of corona discharge or arcing increases with the frequency. Even if the arcing is insufficient to puncture the crystal, its presence will cause either wet or dry oxides to form on the crystal and the electrodes, thereby reducing the activity and greatly shorten- ing the crystal's useful lifetime, if not preventing its operation entirely. Many factors contribute to the possibility of a breakdown: type of holder, presence of sharp edges, smoothness and parallel-- ism of crystal and electrode faces, type of cut, air- gap dimensions, d-c and r-f voltages, frequency, and the like. However, since the arcing in all cases is the direct result of ionization of the air between the electrodes and the crystal, this danger may be removed if the more vulnerable air spaces are filled with an elastic cushion that has little tendency to ionize. It is necessary that the material have a dielectric constant much higher than that of air, and preferably higher than that of the crystal, and that it have low dielectric losses at the operat- ing frequency; otherwise the special advantages of the particular types of mounting with which dielectric material is used would be destroyed by an increase in damping. The dielectric filler may consist of insulating sheets cut to fit a particular mounting, or it may be coated over the electrode faces. In either case, a material of high dielectric constant will permit a crystal to be driven near its elastic limit without the danger of corona ef- fects, and with much less restraint of the normal vibrations than occurs when the crystal is in direct contact with the electrode faces. Suitable dielectric materials are mica, thin sheets of glass or fused quartz or other ceramics, “Cellophane,” nonsul- WADC TR 56-156 56 END VIEW SIDE VIEW Figur* 1-74. Centar-prassuiw typ* til dM»etric tandwich for mounting crystals furous rubber sheeting, cellulose esters and ethers, varnishes, lacquers, vitreous enamels, metallic oxides, rubber coatings applied by electro-deposi- tion, rubber containing resin and other fillings, or fused coatings of natural or synthetic resins. The sheets or coatings should be from 1 to 5 mils in thickness, but care should be taken that the thick- ness of the insulating material does not approach a multiple of a half-wavelength of the acoustic waves th^t will be generated. In any event, the addition of the dielectric material will tend to raise the impedance and frequency slightly, so that in extreme cases it may be necessary to grind the crystal to a frequency lower than that at which it is to operate. 1-144. Figures 1-78 to 1-77 indicate different methods in which the dielectric fillers may be used in mounting a crystal. Figure 1-73 illustrates a pressure tjTpe of mounting with two sheets of di- electric material — arnica, for instance — inserted be- tween each electrode and crystal. Note that the mica extends beyond the edges of both electrodes. This feature is important, for although in a well- designed pressure sandwich no air spaces exist between the crystal and electrode faces, so that ionization and arcing do not occur at the major surfaces, corona discharges can and do occur at the edges, particularly if the sides of the chamber are close in, as is usual, causing the alternating field around the edges to be more intense. How- ever, with the insulating sheet of high dielectric constant overlapping the electrode edges, the in- tensity of the electric field will be greatly dimin- ished. That part of the dielectric directly between the crystal and the relatively inelastic electrode, acts as an elastic cushion, and thus serves much the same function as an air gap, but without in- creasing the possibility of corona or arcing effects. 1-145. Figure 1-74 shows a top and a side view of a center-pressure typo of mounting, where two circular electrodes of small cross section are sepa- rated from the crystal faces by insulating sheets Section I FabricaHen of Crystal Units of high dielectric constant. Again, it is important that the insulation extend well beyond the edges of the electrodes. This arrangement increases the length of the shortest possible arcing path, and, in so doing, diminishes the chance of the occurrence of a discharge. 1-146. Figure 1-75 illustrates two methods by which a comer-clamped air-gap holder can be con- verted into a dielectric sandwich while still retain- ing the principal advantages of the air-gap mount- ing. Figure 1-75A shows an insulating sheet cut to the dimensions of the air gap, and figure 1-76B shows a corner view of the assembled sandwich. Figure 1-75C is a side view of a similar sandwich. (C) FIguro 1-75. Two mothodt (B and C) by which a tornor-elampod, air-gap holdof can bo convortod into a diolottrle tandwieh. FIguro A ahowt a dioloctrle thoot cut to tho dimensions of tho air gap WAOC TR 56-156 57 Section I Fabrication of Crystal Units Flgun 1-76. Oieioctric sboot cut to Ml air gap at nodal-elampad mounHag but with two additional insulating sheets inserted to cushion the crystal entirely from direct contact with the electrode risers. 1-147. Figure 1-76 illustrates the cut of an insu- lating sheet for converting a nodal-clamped, air- gap mounting into a dielectric sandwich. Two niches in the edges of the sheet are cut to fit the two risers of an electrode. When assembled, the sandwich is similar to the comer-clamped model of figure 1-75B ; or, if additional rectangular sheets are inserted next to the crystal, the assembly will resemble the arrangement in 1-76C. With either method, maximum rigidity is obtained for the nodal mounting with a minimum in damping. 1-148. Figure 1-77 is a cross-sectional view of a gravity type air-gap mounting with the electrodes coated with an insulating material of high dielec- tric constant. It is characteristic of air-gap holders that the smaller the thickness of the air gap, the higher the r-f voltage that can be applied before arcing occurs between the crystal and electrode faced. When the electrodes make perfect contact with the crystal, not only are the opposing sur- faces theoretically at the same potential, but no ionizable substance lies between them in which an arc can form. However, the introduction of an air gap not only inherently reduces the electrome- chanical coupling of a crystal unit, but also effec- tively lowers the voltage that can be practicably applied. To remove the latter restriction without diminishing the advantages the air gap provides, the arrangement shown in figure 1-77 can be used. Note that the coating covers the edges of the elec- trodes — an important consideration since it is at the points of sharpest curvature that ionization is most likely to arise. 1-149. The use of insulating sheets and coatings of high dielectric constant permits a crystal to be operated near its elastic limit without the danger WADC TR 56-156 59 figuia 1-77. Cross-socf/onol vi»w of gravltf-air-gap mounfing with olactrodo surfacos protottad by coating of high-dioloctrk matorial of arcing, and hence this type of crystal unit can be operated at higher drive levels than would otherwise be possible. The dielectric sandwich would be advantageous in filter circuits where high amplitude signals are to be encountered; or in small portable transmitters where several ampli- fier stages are not possible, and the excitation level must be as high as possible ; or in any radio trans- mitter designed to be keyed in the oscillator stage where it is important that the oscillations built to peak amplitude in a minimum number of cycles. The insertion of the dielectric sheets also improves the stability and selectivity of the sandwich-type holders, inasmuch as they eliminate direct contact between the crystal and the relatively inelastic electrodes. The principal disadvantages of the di- electric sandwich are the reduced piezoelectric coupling caused by the separation of the electrodes from the crystal, and the damping effect of the frictional and small dielectric losses which are slightly greater than those of the air — provided the crystal is operated well below its elastic limit. notod-DloIcctric Sandwich 1-160. This type of mounting is essentially the same as the previously described dielectric sand- wich except that a thin layer of conducting mate- rial is interposed between the dielectric sheets and the electrodes, or between the dielectric sheets and the crystal, or both. The conductive surfaces may be strips of metal foil not more than 1 mil in thick- ness, or they may be plated, painted, or sprayed directly on the insulating material. Suitable con- ducting substances are copper, nickel, silver, gold, platinum, and their alloys. The conducting layer may be coated on one or both major surfaces of the insulating sheet, or it may completely cover the edges as well as the major surfaces, thus effee-. tively converting the sheet into a highly compliant metal plate. 1-151. Figure 1-78A illustrates the corner-clamped- air-gap mounting using dielectric plates having conducting films on both major surfaces. The two t ftgun 1-7B. (A) Air-gap mounting u§ing dMattrk shoots having conducting Alms on ooch major surloco. (B) Air-gap mounting using dlo/octrlc shoots having a tondueting ft/m on major suifoco In contact with o/octrodo. /Cl Dioloetrk sandwich In which loavos of motal foil aro Insortod botwoon dioloetrle and Cfystol films that are in direct contact with the electrode risers prevent the establishment of differences of potential across the air gaps, and hence obviate the possibility of arcing or corona discharges in these spaces. Figure 1-78B illustrates the nodal-clamped, air-gap mounting using dielectric plates having a conducting film on only one surface. In the case of air-gap mountings, if only one conducting sur- face is to be interposed between a crystal and each electrode, it is preferable that this surface make contact with the electrode rather than the crystal, so that possible electric fields will be “shorted” around the air gap. Figure 1-78C illustrates a di- electric sandwich mounting in which leaves of metal foil are inserted between the dielectric plates and the crystal. The metal foil, being very thin and flexible, snugly fits the crystal surface and in- terferes but little with the crystal’s vibrations. On the other hand, its presence insures a uniform po- tential at all points on the crystal's surface, thus protecting the surface from the effects of exces- sive electric stresses. 1-152. The plated-dielectric sandwich combines the advantages of the plain dielectric sandwich with improvements in the frequency stability, crystal life, frequency spectrum, and piezoelectric coupling. The improvement in frequency stability is greatest in the case of the air-gap crystal units, for the danger of arcing or corona discharge in an air gap is removed without the insertion of a di- electric sheet to fill the gap. Since the damping effect of the air is less than that of the insulating material, the use of conducting film permits a Soctien I / Fabrication of Crystal Units closer approach to the high selectivity of the pure air-gap mounting for crystals which are to be driven near their elastic limit. 'The insertion of a metallic film next to the crystal surface serves to reduce possible electrical stresses at the surface which might indirectly aid the production of small fractures, or cause ionization and chemical effects that would lead to a weathering of the crystal’s face. The insurance of a uniform potential at all points on the surface of the crystal also improves the frequency spectrum, particularly at very high frequencies, where many possible overtone modes can vibrate at frequencies close to that of the de- sired mode. However, the majority of the unwanted modes will have changes of phase and differences in amplitude along the major plane of the crystal, so that the resulting eddy currents that they in- duce in the conducting surfaces will aid in damp- ing them out. Where the interfering modes might otherwise lead to a frequency drift or jump, the damping effect will be reflected principally as an increase in the effective electrical resistance and as a decrease in activity. Finally, closer piezoelec- tric coupling is achieved if the entire insulating material is given a metallic coating. The dielectric sheet thus effectively becomes an extension of the electrodes, and the close coupling of the simple sandwich mounting is approached, but without the heavy damping caused by friction between the crystal and solid metal. Battan Manats 1-153. The ceramic button crystal mount repre- sents the ultimate in crystal-holder design yet to be reached via the sandwich and air-gap evolu- tionary chain. Originally, the button electrode was developed as an all-metal modification of the cor- ner-clamped, air-gap type. As illustrated in figure 1-79A, the all-met^ electrode is provided with conventional lands at the corners, but the effective center area has been reduced by surrounding the center with a relatively deep circular groove. The effect is to reduce the shunt capacitance across the crystal while retaining a central area of sufficient size for adequate excitation; the reduction in shunt capacitance is particularly desirable if the crystal is to be operated in the v-h-f range. Also, since the principal excitation is confined to a cen- tral circular area, the likelihood of spurious modes is somewhat reduced, because the vibrating part of the crystal tends to exhibit the properties of a circular plate. The superior frequency spectrum of the circular plate is probably even more closely ap- proached by using electrodes having solid ring- shaped lands that completely surround the circular WAOC TR 56-156 59 Section i Fabrication of Crystal Unite (A) (B) ALL-METAL BUTTON ELECTRODE (C) I CRYSTAL HOLDER HC-IO/U ASSEMBLED UNtT ALL-METAL SOLID -LAND ELECTRODE NICKEL- SILVER BARREL SPRING ASSEMBLY AND CONTACT- MOUNTED HERMETIC SEAL CONTACT- MOUNTED HERMETIC SEAL CERAMIC -BUTTON ELECTRODES FIgun 1-79. Button electrodos and mmthodt of mounting air gap at the center. See figure 1-79B. However, it is by combining the advantages of the button mounting with those of the plated-dielectric sand- wich and circular quartz plates that optimum per- formance is obtained for thickness-shear modes at very high frequencies. Figure 1-79C shows the principal parts and the complete assembly of Crys- tal Holder HC-IO/U. The shunt capacitance is held to a minimum, first, by the use of ceramic sup- porting plates in place of all-metal electrodes, and second, by the use of a coaxial electrode system in place of the conventional method where the crystal leads parallel each other through the base assembly. The ceramic-button electrodes are usu- ally very thin metallic platings that cover a small circular area at the center of each ceramic plate. Although the lands may be provided by forming thickened sections at the rim of the ceramic disks, usually they are obtained by plating metal risers on the surface of the ceramic; these plated risers are not connected electrically to the center metallic section. The air-gap thickness is normally between three and five microns. A notch in each ceramic button permits an extension of the electrode plat- ing to the opposite side, so that contact with the crystal leads can be made with a minimum of in- crease in electrode capacitance. This type of crys- tal holder is unequalled in performance when used with harmonic-mode crystals in the very-high-fre- quency range. It should be noted, however, that one of the original advantages of the plated-dielec- tric sandwich mounting is not fully realized in the case of ceramic-button electrodes — namely, the protection against arcing or corona discharges. For this reason, the ceramic-button crystal units will not withstand as high a drive level as might other- wise be possible. On the other hand, the thin air gap that can be obtained, the presence of a high- dielectric material almost flush with the edges of the plated electrodes, and the firm mechanical sup- port by which the crystal is held and cushioned against shock make this unit more durable under high drive levels than conventional air-gap holders. One of the more important advantages of the ceramic-button is the reduction of spurious modes through the use of circular quartz plates and small electrodes. The small electrode dimensions serve to concentrate the activity at the center of the crys- tal, where the crystal is most likely to be of uni- form thickness; thus, sudden frequency jumps are prevented, for these seem to be due primarily to abrupt shifting of the center of activity between areas having slightly different average thicknesses. Plated Crystals 1-154. Since 1940 the designers of crystal units have increasingly favored the use of electrodes in the form of extremely thin metal films deposited directly on the crystal. Coatings of silver and gold have been successfully applied by spraying and baking, but in general the most advantageous method is by evaporating the metal in a vacuum and allowing it to condense on the exposed sur- faces of the crystal. Sputtering processes are being used increasingly, particularly for base plating, where the crystal is plated in vacuum by ionic bombardment from high-voltage negative elec- trodes composed of the desired plating metal. Elec- troplating of crystals also finds application. The noble metals, gold and silver, are the elements most commonly used in plating crystals because of their resistance to oxidation, their relative ease of plating, and the strength of their soldered junc- tions. Other metals that are used in plating are nickel and copper. Aluminum plating is preferred if a crystal is to be held in position by pressure pins or knife-edge clamps. This is because alumi- num is more durable to frictional wear, and be- cause its lesser density permits an electrode of lighter weight. However, alumimun is the more difficult to apply, has a tendency to oxidize, and its soldered connections are not as strong as those of WADC TR 56-156 60 ( silver or gold. For these reasons, silver is more widely used if the crystal is to be soldered between wire supports, and gold is used if the wire-sup- ported unit requires maximum stability and resistance to aging. Aluminum coatings are com- monly applied at 1 milligram per square inch, which is equivalent to a thickness of approxi- mately 0.0225 mil; silver is applied at 4 milligrams per square inch, a thickness of approximately 0.0232 mil ; and gold is applied at 3 milligrams per square inch, a thickness of approximately 0.0114 mil. The actual plating procedure may be divided into two or more steps involving more than one plating process. As an example, the Signal Corps Engineering Report E-1108 by J. M. Roman rec- ommends as many as three different plating stages during the fabrication of low-resistance, 50-mc harmonic-mode crystal units of the CR-23/U type. The base plating is accomplished by a sputtering machine in which a group of crystal blanks are mounted in a rectangular metallic mask midway between two gold electrodes, which are 51/2 inches square and inches apart. A bell jar is placed over the electrode assembly and is evacuated to 0.05 to 0.02 millimeters of mercury. 2200 volts dc are applied between the crystal mask and the elec- trodes; first for 30 minutes at 100 ma with the mask negative in order to clean the crystals by ionic bombardment, and next, for 37 minutes at 100 ma with the electrodes negative for the actual gold plating operation. A second sputtering ma- chine is used to clean the crystal mask of the gold deposited upon it during the plating procedure. This latter operation requires an hour at 100 ma. After being mechanically mounted on HC-6/U bases between 9-mil, edge-clamping spring wires, the crystals are given a preliminary performance test. If a crystal is more than 0.1 per cent off its nominal frequency it is subjected to an additional plating process, "niis time it is plated electrolyti- cally with nickel. The electroplating of 50-mc crystals proceeds at a rate of 0.9 ma, which is equivalent to a harmonic frequency change of 100 kc per minute. The electrolytic solution consists of chemically pure nickel ammonium sulphate, boric acid, ammonium chloride, and water in a weight ratio, respectively, of 75/16/15/1000. After mount- ing, testing, and electrically bonding the plated crystal electrodes to the supporting wires with sil- ver cement, the crystals are given a final spot plating with gold in an evaporation type plater to bring them to their specified frequency. This final plating process is accomplished in vacuum while the crystal is connected in a test oscillator circuit. 1-155. The advantages of using metal-film elec- SecHen I Fabrication of Crystal Units trodes are several fold: maximum piezoelectric coupling is achieved ; the possibility of arcing be- tween the electrodes and the crystal is reduced to a minimum ; variations of frequency due to a shift of the position of the crystal relative to the elec- trodes are eliminated; frictional losses and wear due to inelastic contact between the crystal and the electrodes are removed; the metallic film aids in protecting the crystal from erosion; the film is readily adaptable for various types of nodal mount- ing, and is easily divided into several electrodes for use in exciting particular harmonic modes. All in all, the plated crystal is the most practical for obtaining optimum crystal performance at low and fundamental-mode high frequencies. The metal-film electrodes, however, have certain dis- advantages; the metal has a tendency to absorb moisture, thereby causing the frequency to change ; when clamp supports are used, friction at the clamped points will eventually wear away the metal coating; and generally, the mounting tech- niques are more critical for plated crystals. Pressure-Pin Mounting 1-156. Pressure-pin holders (see figure 1-80) are used to support low-frequency (up to 200 kc), electrode-plated crystals, particularly those crys- tals used in telephone filters. Each crystal is clamped at the center of a nodal zone by one or more pairs of opposing pins. For crystal plates one-half inch square and smaller, the diameter of the pins is approximately 10 mils, and the clamp- ing force varies from one to two pounds ; for larger plates, pins of larger dimensions exerting some- what greater clamping forces are used. It is im- portant that the plated electrode be of aluminum, for the greater hardness of aluminum is required to resist the wear at the points of contact with the pins. Normally, these holders are designed for mounting a complete set of filter crystals within Figun 1-80. Pressur 0 -pin mounting, with provltioni for supporting four platod filtor trystals * WADC TR 56-156 61 Section I Fabrication of Crystal Units a single hermetically sealed container. The holder shown in figure 1-80 mounts four crystal elements. 'I’he pressure is applied by the springs mounted at the ends, and the pins serve as electrical connec- tions as well as mechanical supports. For greater mechanical stability, slight niches may be made in the quartz at the clamped points. 1-157. The pressure-pin holder has the advantages and disadvantages of the plated electrodes, and is used primarily for low- and medium-frequency filter crystals. It is particularly applicable for use with face-shear elements, since these have but one practicable nodal spot for clamping. The chief limitation of the pressure-pin mounting is the mechanical impedance it introduces. If the diam- eter of the pin is made too small, the crystal will tend to rotate about its axis of support; however, the larger the diameter is made, the more the clamping area will extend beyond the nodal point. To obtain optimum performance with this type of mechanical support, a resonant-cantilever de- sign foi’ pressure pins was invented by J. M. Wolfskin of the Bliley Elect’dc Company (U.S. Patent 2,240,453, 1941). This step was quite sig- nificant, not only in its own right, but because it provided a forerunner of the resonant-wire type of mounting. The following discussion is based on an analysis of the cantilever clamp by R. A. Sykes (Bibliography No. 741). The Cantilever Clamp 1-158. The cantilever clamp is a pressure-pin sup- port designed to resonate at or near the crystal frequency. Figure 1-81 illustrates a pin mounted as a cantilever, and figure 1-82 indicates the motion of the cantilever as a quarter-wavelength bar with a node at the fixed base and a loop at the point of contact with the crystal. It is important that even quarter-w'avelengths be avoided, else the mechanical energy returning to the crystal will Figure 1-81. Cantilever clamp for providing a resonant-pin support for the crystal * Figure 1-82, Resonant motion of cantilever pin when its length is equal to one-quarter wave-length of clamp-free flexural vibration. Note that the effective free end of the pin is that end supporting the crystal (not shown) * oppose its motion, thereby greatly increasing the impedance and lowering the activity. The length of a cantilever pin that will present a loop to the crystal can be determined approximately from the frequency formula of a clamp-free rod in flexural vibrations : , mMv “ 8ir 1- where: m = 1.875 for the 1st node of vibration of the rod (pin) m = ^n — for n = 2, 3, . . . d = diameter of pin V = velocity of propagation along pin 1 = length of pin For phosphor-bronze pins, v = 3.6 x 10* cm/sec; therefore, to support a 100-kc crystal, pins 1 mm in diameter should be 2.25 mm long to resonate in the mode indicated in figure 1-82. To resonate as a three-quarter-wavelength rod, n = 2, and 1 = 5.67 mm for a pin of 1-mm diameter. The pin should be rounded at the end, as shown in figure 1-81, so that firm contact is made without the risk of having all the clamping force concentrated momentarily at a sharp point. 1-159. A properly designed cantilever clamp should extend the useful range of the pressure-pin type of mounting to somewhat higher frequencies, and this has proved to be true in actual practice; how- ever, at the present time no data is available con- cerning its frequency application above 350 kc. Theoretically, a pair of pins could be used at any of their clamp-free harmonic modes, and thus pins of the same design need not be restricted to use at a single frequency. The principal promise of the cantilever clamp, however, is that it can provide a firm mechanical support while presenting a mini- mum of interference to the normal vibration of the crystal. Knife-Edged Clamp 1-160. The knife-edged clamp is similar to the pressure-pin method of mounting, except that the WADC TR 56-156 62 Section I Fabrication of Crystal Units QUARTZ CRYSTAL PHOSPHOR BRONZE LENGTH i TO PAPER SPRING Figure 1-83. Knife-edge clamp support for two crystal plates. Each crystal has two pairs of plated electrodes, and is so mounted that each pressure blade makes electrical contact with a different elec- trode. This arrangement effectively provides four crystal elements for use in a balanced filler circuit * clamping prongs are blade-shaped, as indicated in figure 1-83. The dimensions of the clamping points are, on the average, about 35 mils in length, and 10 to 15 mils in width. These blades are used with crystal elements that have a nodal axis parallel to the plane of the major faces, and care must be taken to make certain that the blades are centered along the nodal line. Pressure is applied by phos- phor-bronze springs, with the blades serving as electrical connections as well as mechanical sup- ports for the crystal. The holder shown in figure 1-83 mounts two crystals, but, because the plated metal films are divided to provide two electrode pairs for each crystal, the equivalent of four crys- tal plates is effectively available for use in a balanced filter circuit. 1-161. The advantages of the knife-edge clamp are essentially the same as those of the pressure-pin mounting, except that the greater surface of con- tact between the crystal and the clamp permits a firmer mechanical support. However, the knife- edge clamp is limited to use with those crystal elements that have W’ell-defined nodal lines. Its most important application has been as a mounting for the —18"' X-cut filter crystal, a crystal that can vibrate in a very pure length-extensional mode, and which has a nodal axis at the center parallel to the width dimension. The knife-edge clamp is gen- erally useful only at frequencies below 120 kc. Wire Mounting 1-162. Wire-mounted crystal units are of two kinds: those that employ wire supports designed to resonate at the crystal frequency in a manner similar to that described in paragraph 1-158 for cantilever clamps, and those that clamp the crystal at the edges by non-resonant spring wire. This latter type of wire support is the cemenfed-lead mount, which is classified here as an edgevclamped mount. The wire mounting provides a firm but flexible support that serves to cushion the crystal from external vibration and shock. In addition, it can combine the advantages of the metal-film elec- trode with the low impedance of resonant supports, and can be used to mount any of the crystal elements, both high- and low-frequency plates, vibrating in extensional, shear, or flexural modes. Because of these advantages, the wire-type mount- ing is generally favored for crystal units used in military equipment. 1-163. There are two principal types of resonant wire mounts, the solder-cone and the headed-wire. In general, the solder-cone support is restricted to relatively small crystal plates — for example, to frequencies above 300 kc for C elements. The headed-wire type is more suited to larger plates. Solder-Cone Wire Support 1-164. A diagram of the solder-cone type of wire mounting is shown in figure 1-84, and a mounted crystal is shown in figure 1-85. The crystal to be mounted is first spotted with small silver footings, 40 to 90 mils in diameter, at the nodal points where the wires are to be attached. Next, the elec- trodes are plated on the crystal by an evaporation or other process. Silver is generally used, although gold may be preferred where i-esistance to corro- sion is paramount. Aluminum has not been widely used in wire-mounted units, because of the weak junction it makes with the solder. However, recent experiments indicate that an aluminum junction with a solder of indium (a rare, fusible metal, chemically similar to aluminum) is quite strong, so that eventually greater application may be found for aluminum-plated crystals. The mount- ing wires are normally of phosphor bronze, be- cause of its high tensile strength and resistance to fatigue. A eutectic tin-lead solder is used that would normally be an alloy of approximately 63 percent tin and 37 percent lead by weight; how- ever, to prevent an excessive diffusion of silver molecules from the silver spot into the solder dur- ing the soldering operation, the solder should con- tain 0.1 percent silver if the soldering is performed by hot-air blast, or a 59.5 — 34.5 — 6 percent tin- lead-silver combination if performed by hot iron. A solder cone in the shape of a bell (see figure 1-84) has been found to provide the best perform- ance characteristics, and is the type of cone that WADC TR 56-156 63 Section I Fabrication of Crystal Units Figure 1-84. Solder-cone resonanf-wire support. The solder ball "tunes" the wire to the crystal frequency if it is placed at a distance equal to an odd multiple of a quarter wavelength (I,, 1^, etc.) from the peak of the solder cone * is least likely to rupture at the peak to form a "crater.” For small crystals, the part of the wire enclosed by the cone may be straight, but for larger crystals sufficient anchorage requires that the end of the wire form a small hook. The wire is tuned to resonance by fixing the position of a solder ball at an odd quarter-wavelength from the peak of the cone; the solder ball serves as a "clamped” point for reflecting the wave energy back to the crystal. The "free” end of the wire is effectively at the point where it enters the solder cone. The distances h, I™, and I, indicated in figure 1-84 mark "free lengths” of wire that will be reso- nant at the given wavelength. Note that each of the lengths defines a distance from the "free” end of the wire to a node where the solder ball should be placed. 1-165. Theoretically, the resonant lengths 1,, Ij, 1:„ . . . obey the same clamp-free frequency equation that is given for the cantilever clamp in paragraph 1-158. Experiment, however, has demonstrated that somewhat longer lengths are required for optimum performance. Normally, the free length of the wire is made a quarter-wave section, 1,, in the frequency range of 20 to 250 kc, and a three- quarter-wave section, L, in the range of 250 to 1000 kc. For. phosphor-bronze wire, the empirical formulas for these distances are: Figure 1-85. Solder-cone wire mounting of face-shear element where: d = diameter of wire in inches (usually 0.0035, 0.005, 0.006.3, or 0.008 in.), f = frequency in kc. 1-166. After soldering to the crystal, the support- ing wires are bent to make them serve as springs. One, two, or three bends are carefully spaced and directed so that the displacement per unit force will be the same for all directions. The ends of the wires are then soldered without tension to metal rods, or "straights,” which in turn are welded to eyelets staked in a mica or bakelite base. In mounting small crystal plates, the straights are little more than short, metal stubs, but larger crys- tals are mounted in "cages” having a mica roof as well as a mica base. Figure 1-86 shows the cage assembly of a 40-kc length-width flexure crystal. The cage is formed by two mica plates at each end, and four straights. Besides providing for the proper mounting of the straights, the mica plates also serve as “bumpers” for the crystal. The inner and outer plates limit the horizontal and vertical WADC TR 56-156 64 section I Fabrication of Crystal Units displacements respectively, thereby protecting the unit from wire or crystal damage in the event of severe shock or vibration. The spacing between bumper and crystal is normally between 25 and 30 mils. Where the operating frequency is below 3 kc, the wavelength is usually sufficiently long for the entire wire to be cut to a resonant length, so that the soldered junction at the straight can serve as the nodal terminal. However, the optimum free length of wire becomes increasingly critical as the frequency is raised. Solder balls are used to estab- lish resonance, but at low frequencies small metal disks are threaded on the wire to provide greater mass while permitting a precise adjustment to the correct position; after adjustment, the disks are loaded at the back with the correct amount of solder. Better control is obtained at the higher frequencies without the disk. The solder weights range from approximately 80 milligrams for 8-mil wire, for large crystals, to 6 milligrams for 3.5-mil wire, for small crystals. 1-167. The principal disadvantages of the solder- cone wire support arise from the effects of the solder cone upon the electrical characteristics of the crystal. To provide a junction of given me- chanical strength, a certain quantity of solder is required. The solder, however, considerably in- creases the effective resistance of the crystal cir- cuit as the temperature becomes high; if a high crystal Q at high operating temperatures is re- quired, the solder cone must be small, and, conse- quently, the crystal unit cannot be as rugged mechanically as would otherwise be possible. Con- versely, if the crystal unit is to withstand severe mechanical vibrations and high operating tem- peratures, the solder cone must be of maximum size, so that the Q and frequency stability will necessarily be at a minimum. Furthermore, as the volume of solder is increased appreciably, the tem- perature-frequency characteristics of the crystal may be considerably changed. Normally, the tend- ency will be for the zero temperature coefficient to shift to a lower temperature ; in extreme cases, the zero point may be lost altogether. The tempera- ture-frequency effects of hooked wire are generally more pronounced than those of straight wire, when equal volumes of solder are used. Another consideration is the difficulty experienced in mak- ing two cones of the same dimensions. Headed-Wire Support 1-168. The headed-wire support (see figure 1-87) was developed to obviate the disadvantages of the solder cone, while preserving all the advantages of the wire type of mounting. The head of the wire, which resembles that of a common pin, has a diameter of approximately 22 mils for 6-mil wire. It is pretinned, and a small globule of solder is left at the end for sweating to the crystal ; the FACE FLEXUAE Figun 1-86. Capo assembly for soldor- cone wire mownfinp of low-fraquonty fongth’Width flexure crystal * volume of solder varies from 1000 to 7000 cubic mils, according to the size of the crystal. Phos- phor-bronze wire is used, and all other mounting details are substantially the same as those for the solder-cone type of support. 1-169. The headed-wire is superior to the solder- cone mounting, because it provides a greater and more uniformly distributed mechanical support with a smaller quantity of solder; in the case of low-frequency crystals, the Q is improved by as much as twenty-five percent. Furthermore, the Figure 1-87. Headed-wire crystal support * WADC TR 56-156 65 Section I Fabrication of Crystal Units distance d (figure 1-87) is a constant for all crys- tal units of the same design, so that the resonant free length of the wire can be predetermined ac- curately, thus permitting smaller tolerance in the rated characteristics. An additional advantage is that the headed wire diminishes the mechanical coupling between the vibrating systems repre- sented by the crystal and the wires. Standing waves are caused, not only by reflections between solder ball and crystal, but also to a certain extent by reflections from one solder ball, through the crystal, to the solder ball on the opposite side. By reducing the coupling between crystal and wires, the impedance effects due to the interfering through-waves are reduced, and a purer frequency spectrum is possible. Headed wire may be used to replace any other type of low- and medium-fre- quency crystal mounting, and a well-designed headed-wire crystal unit will generally surpass the other types in all-round performance. However, at the higher frequencies a clamped air-gap holder is still to be preferred for greater activity and frequency stability, and at low frequencies, ulti- mately the cantilever clamp may prove superior for genera] use. Edge-Clamped Mounts 1-170. Two variations of the edge-clamped type of mounting are illustrated in figure 1-88. The mount EFFECTIVE ELECTRODE Figure 7-88. Cdge-clamped systems of mounting. (A) Mounting for low-frequeney crystals. (B) Cemented- lead mounting for high-frequency crystals shown in (A) has been used with low-frequency crystals vibrating in extensional or flexure modes ; the mount in (B), known as the cemented-lead mount, is widely used as an alternate to air-gap holders in mounting high frequency, thickness- shear elements. Although edge-clamped mounts have been successfully used in the production of high-activity crystal units for both high- and low- frequency applications, this type of mounting when used with low-frequency crystals, is probably somewhat inferior to well-constructed headed-wire or resonant-pin supports. However, a special fea- ture of the edge-clamped mounting system is the method of dimensioning the electrodes (a method also adaptable for use with resonant pins), by which optimum performance characteristics can be obtained with high-frequency crystals. Plated electrodes (or metal foil cemented to the crystal) are used, but, as shown in figure 1-88 (B) , the crys- tal faces are only partially plated, and the plating on opposite faces is extended to opposite edges only, so that the effective electrode area is concen- trated within a small circular region at the center of the crystal. By this means the capacitance is kept small, and the principal activity is confined to the central region, where the crystal is most likely to be of uniform thickness. Both of these factors are advantageous in improving the frequency sta- bility. Also by reducing the activity in the vicinity of the edges, much of the damping due to the im- pedances of the supporting structure is obviated. Mechanical support and electrical connection is supplied by tinned, high-quality spring piano wire, which is clamped and cemented to the crystal at the edge where electrical contact can be made with the lead-outs from the electrodes. The cementing is used principally for the purpose of insuring good electrical connection, and not for supplying mechanical support, which should be provided by spring-wire clamps. The base ends of the spring wire are coiled around and soldered or welded to the base stubs. Although the supporting wires are not designed to be resonant elements of the crystal unit, they do provide the protection against shock and external vibration afforded by the other types of wire mounting. As compared with the per- formance of fundamental-mode, thickness-shear crystals, such as elements A and B, that are mounted in corner-clamped air-gap holders, the performance of the same elements, when wire- mounted, will generally be superior. In addition, the wire mounting permits the use of smaller crystal holders. The elimination of the air gap re- duces the likelihood of arcing, but this does not mean that the wire-mounted units can be operated WADC TR 56-156 66 at higher voltages than the conventional air-gap crystal units. This is because the wire-mounted crystal is more isolated thermally and tends to be- come hotter. The advantages of the cemented-lead over the ceramic-button mounting system are less pronounced than the advantages over the other air-gap systems. For operation at frequencies from 1 to 10 me, the wire-supported crystal usually has the better operating characteristics. As the fre- quency increases, however, the metal plating of the wire-mounted element becomes an increasingly greater factor in damping the oscillations ; and in the upper very-high-frequency range, above 100 me, non-plated crystals that are pressure-mounted between ceramic buttons are definitely to be pre- ferred. Even in the fundamental frequency range, the ceramic-button mounts, which provide the better mechanical protection, may be used with good effect, and optimum performance character- istics for given operating conditions might better be achieved by combining the merits of wire- mounted edge clamps with those of plated dielec- tric buttons. HOUSING OF CRYSTAL UNITS 1-171. The principal function of the housing is to provide a hermetically sealed, moisture-resistant container. Plastic housings of sandwich, air-gap, and clamp-type holders are normally sealed with neoprene gaskets. Natural rubber is not recom- mended, as the sulphur used in processing the rubber will ultimately contaminate other parts of the holder. Wire-mounted units are readily adapt- able for housing in metal or glass tubes, employing standard radio parts; however, small, two-pin holders are generally preferred. Before sealing, a crystal unit is exposed to high temperature in an evacuated oven, in order to drive off adsorbed gasses. The sealing itself is usually performed in dry air, although certain crystals, particularly the flexure-elements, are sealed in vacuum. Optimum performance is obtained when a crystal is mounted in an evacuated container, since the damping effect of the air is eliminated. 1-172. If metal, rather than glass, housing is em- ployed, it is difficult and expensive to seal a Crystal unit so perfectly that not even minute leaks will develop due to stresses on the pins and the glass- sealing of the eyelfts. For this reason most crystal units are sealed in dry air, so that if very small leaks are present, the crystal characteristics will not be appreciably affected for a long period of time. Leakage is minimized if the base is rigidly protected against deformation, and if the glass ^ Section I Fabrication of Crystal Units sealing fills the entire eyelet cavity uniformly. However, if a crystal is to be mounted in vacuum, a glass housing is to be preferred. AGING OF CRYSTAL UNITS 1-173. “Aging” is a general term applying to any cumulative process which contributes to the de- terioration of a crystal unit and which results in a gradual change in its operating characteristics. There are, of course, many interrelated factors in- volved in aging — minute leakage through the con- tainer, adsorption of moisture, corrosion of the electrodes, ionization of the air within the con- tainer, wire fatigue, frictional wear, spurious elec- trolytic processes, small irreversible alterations in the crystal lattice, outgassing of the materials com- posing the unit, over-drive, presence of foreign matter, various thermal effects, pin strain due to socket stresses, and erosion of the surface of the crystal. However, if a crystal unit is well designed and carefully constructed, the rated operating characteristics may well outlast the equipment in which the crystal is used. 1-174. Usually the first effects of aging can be traced to changes at the surface of the crystal. These changes may be due directly or indirectly to almost any combination of the factors mentioned in paragraph 1-173, and their occurrence can be avoided or greatly diminished only if proper pre- cautions and techniques are employed during manufacture, and if low driving voltages are em- ployed during operation. To produce a crystal unit of long life, the final stages of production require particular precautions. These concern the finishing processes of lapping, etching, cleaning, mounting, heat cycling, and protecting against moisture. Lopping to Reduce Aging 1-175. Whether a crystal is being ground with abrasives which are cemented or imbedded in a grinding disk, or lapped with loose abrasives under a lapping disk, the cutting proceeds by virtue of the small fractures and chips which result when the hard, sharp edges of the abrasive particles are rubbed against the surface of the crystal. Com- mercial crystals are usually produced by lapping with loose abrasives, instead of grinding by “grindstones,” except in the initial cutting stages and the final edging process, where diamond saws are commonly used. Each succeeding lapping stage employs a finer grade of abrasive, and must com- pletely remove the surface left by the preceding stage. The final lapping requires very fine abrasive particles, such as 1000- to 1200-grain carborun- dum, and should preferably be performed in a WADC TR 56-156 67 Section I Fabrication of Crystal Units mixture of abrasive, castile soap, and water. To reduce aging, soap and water are preferred as the coolant in the finishing stage, rather than kerosene or other oils, although kerosene permits a faster cutting rate for the same abrasive and lapping speed. Apparently, the residue of fractures re- maining after a soap-water-abrasive lapping does not penetrate as deeply as that remaining after a kerosene-abrasive lapping. Regardless of how fine the abrasive, small fractures and cracks will be left in the surface of the crystal after the final lapping, and in time these cracks will spread, absorb mois- ture, and ultimately result in a weathering of the surface. Additional care must be taken to ensure Figure i-89. Minimum change in frequency that AT and BT plates must undergo due to etching, if the etching is to be sufficient to remove all surface cracks and fissions remaining from the final lapping stage. Note that, as the crystal becomes thinner, a given change in the thickness dimension means a greater change in the frequency. The frequency change for a BT cut is less than that for an AT cut of the same initial frequency, since the larger frequency constant of the BT cut permits a thicker plate that the crystal is not finished with slight con- cavities in the surfaces, or with one end lapped down more than the other, making the crystal wedge-shaped. Although optimum performance is to be obtained with perfectly planar surfaces, greater insurance against unwanted non-parallel- isms is gained if the lapping is controlled to give the plates a symmetrical convex contour of ap- proximately 5 microns for lower-frequency crys- tals, and approximately lO’/f (cycles) microns for crystals above 3 me. Etching to Reduce Aging 1-176. After the final lapping stage, the crystal is normally given an etching bath to remove all for- eign particles. An eight-minute bath in forty-seven percent hydrofluoric acid is sufficient for the aver- age crystal, and will permit a firm contact between the crystal and its electrode coating. An etching time of at least thirty minutes is necessary, how- ever, if a minimum aging and a maximum Q, sta- bility, and drive level are desired. The longer etch- ing period is required to ensure that the deeper fissions in the surface caused by the final lapping are thoroughly removed. However, the deep etch is difficult to control, and particular care must be exercised if the desired dimensions of the crystal are to be achieved. It is customary to divide the deep-etching process into two, or more, steps; ( 1 ) to etch the crystal to within 1 kc of the desired frequency: and (2) in the succeeding steps, to bring the crystal within its tolerance limits. Figure 1-89 indicates the degree of etching required to prevent aging in AT and BT cuts. CluanlinuM to Ruducu Aging 1-177. The protection of a crystal from foreign matter and moisture is of paramount importance if the crystal is to operate with stability and long life. Only minute traces of dirt, dust, or finger- prints on a crystal will cause the performance to be erratic. Cleanliness is necessary throughout the final production period, but particular emphasis is required during the stages immediately prior to sealing. Before and after etching, each crystal blank should be scrubbed thoroughly in soap, or trisodium phosphate, and water with a soft brush ; rinsed in 0.5 percent ammonium hydroxide solu- tion, and again washed thoroughly in running water; dried in an oven heated to 100 degrees centigrade, or in a warm, clean, air stream ; washed again in distilled carbon tetrachloride or other solvent; rinsed in hot distilled water; and finally, carefully dried in an oven. The electrodes and holder must be similarly cleaned, and neoprene WADC TR 56-156 68 tweezers should be used in handling the parts dur- ing the hnal assembly. If the crystal is to be metal- plated, the complete mounting must be cleaned again before sealing. A hot spray of distilled tri- chloroethylene for one-half minute is sufficient. The plated crystal will normally require a small amount of edge-grinding with fine emery paper to bring the mounted unit to the proper frequency. This step unfortunately weakens the aging resistance of the treated surfaces at a stage when further etch- ing is no longer feasible for commercial crystals. However, before testing and sealing, a retouched crystal unit should be thoroughly washed and scrubbed, with every precaution taken to ensure that no foreign matter remains on the crystal or mounting. Where the facilities are available, clean- ing can be performed by exciting the bath with supersonic acoustic waves, which can clean the crystal by shaking all loose fragments off its sur- face. In fact, a supersonic bath can be quite as effective as an etching bath in reducing aging. MouNtiag to Reduce A9in9 1-178. As a general rule, any deviation in the mounting which causes an increase in the fric- tional losses will shorten the useful life of a crystal unit. Thus, in the nodal types of mounting, small deviations from the nodal point in the position at which a crystal is held will shorten the life of the crystal. Wire-mounted crystals require additional precautions during fabrication to avoid local changes or stresses at the surface of the crystal. Particular care must be taken to avoid electrical “twinning," which will occur if the temperature is raised above the inversion point, 573°C, and then lowered again ; or twinning may be induced at a much lower temperature if a sharp temperature gradient is present in the crystal. These precau- tions are necessary during the baking of the silver spots, the division of the electrode coating by elec- tric stylus, and the soldering operation. In baking the silver spots, the temperature should be kept forty to fifty degrees centrigrade below the in- version point, and care must be taken to make certain that the crystals are heated uniformly. Some twinning is inevitable when using an electric stylus to divide an electrode coating; however, if straight-line division is required, the twinning may be avoided by using an abrasive tool or sand blasting in place of the stylus. To avoid thermal stresses during the soldering operation, a heated support should be provided for heating the crystal uniformly to a temperature of approximately 100°C. Twinning, regardless of its cause, primarily affects the steady-state electrical characteristics of Section I Fabrication of Crystal Units a crystal element, and only indirectly contributes to gradual changes in the performance of the crys- tal. At least, no statistical data has been collected to show a correlation between twinning and aging; nevertheless, a series of small twinned spots at the surface is likely to make the area more susceptible to erosion. Readjustments of the crystal lattice at the twinning boundaries after long periods of elec- trical, mechanical, and thermal stresses might be expected ; but if these are due to occur, they can probably be made to take place by a process of artificial aging before the crystal is placed into operation. Twinning, however, raises the induc- tance and effective resistance of an element, and hence, decreases its activity for a given operating voltage. Since the ultimate requirement of a higher operating voltage can lead to a shortening of the life of the crystal unit, an undue amount of twin- ning indirectly becomes a factor in the aging. Twinning will also raise or lower the frequency, according to the particular type of element. If the twinning is introduced during the final stages, this may require a substantial amount of edge-grinding during the final frequency-adjustment stage, and more of the etched surface may need to be re- moved than otherwise. Thus, although "heat” twinning is considered primarily in connection with its immediate effect upon the characteristics of the crystal, it should also be avoided as an in- direct factor in aging. A more direct factor in shortening the life of a wire-mounted crystal unit is a nonuniformity in the soldered junction, which is more likely to occur in a solder-cone than in a headed-wire support. When the stresses are un- evenly distributed, the soldered junction itself will tend to age ; and even if mechanical breakage does not occur, the changes in the electrical characteris- tics will lead to poor performance and instability. Special care must be taken to make certain that the silver spots are of uniform density. The con- tainers of liquid silver should be agitated for sev- eral hours immediately prior to application. Also, the critical nature of the soldering operation re- quires the aid of a machine and accessories of special design. Heat Cycling to Reduce Aging 1-179. A newly mounted crystal will normally ap- pear to age more rapidly than one that has been in operation for a long period of time. This effect is not due to an actual deterioration of the crystal unit, but merely to an initial adjustment of the crystal, particularly at its surface, to its operating environment and changes in temperature. The stabilization period can be reduced to one of very WADC TR 56-156 69 Section I Electrical Parameters of Crystal Units short duration by subjecting the crystal unit to a surface of the crystal and the corrosion of the series of slow heating and cooling cycles varying electrodes. For optimum performance and long between 24°C and 116°C. Metal-plated elements life, every precaution must be taken to ensure that are frequently heat-cycled during the final fre- the interior of the sealed crystal unit is as free as quency-adjustment period, and again after sealing. possible from moisture. Prior to sealing, all com- In a series of tests at the Hunt Corporation, it ponents of the crystal unit should be heated in was found that negative aging (frequency de- vacuum to drive off absorbed water vapor and creases with time) is generally due to insufficient other gases; and if the sealing is performed in cleaning of the crystal unit. When this was rem- air, the atmosphere should not have a relative edied, it was found that the crystal units would humidity higher than 5 percent, then age positively. The cause of the positive aging was traced to the outgassing of the metal Low Drive Level to Reduce Aging plating of the crystal, and its elimination has been 1-181. As a general rule, the lower the drive level, achieved by pre-aging the plated crystal for 3 the longer will be the useful life of a crystal unit, minutes in a 300°C oven. After a sufficient period This is true because the cumulative effects of al- of artificial aging, a properly fabricated and oper- most all of the previously discussed aging factors ated crystal unit will maintain its final tempera- are considerably more pronounced when the crys- ture-frequency characteristics indefinitely. tal is operated at high drive levels. Also, the higher Low Relative Humidity to Reduce Aging operating voltages greatly increase the tendency 1-180. A low relative humidity is of paramount toward corona discharge and other ionization importance if excessive aging is to be prevented. effects, and the vibrations of greater amplitude are Even if a crystal is perfectly mounted and clean, more likely to result in crystal or wire fatigue. To an ambient relative humidity higher than 40 per- ensure maximum lifetime, a piezoelectric reson- cent will sharply increase the insulation resistance, ator should be driven at the lowest practicable and will greatly accelerate the weathering of the level consistent with the circuit requirements. ELECTRICAL PARAMETERS OF CRYSTAL UNITS EQUIVALENT CIRCUIT Or CRYSTAL UNIT trode parts that extend beyond the quartz. Ch. and Ch 2 represent the distributed capacitance of the 1-182. A crystal unit may be represented by the crystal circuit to the holder H. Ca represents the equivalent electrical circuit shown in figure 1-90. capacitance between the electrodes and the crystal Ri represents the terminal-to-terminal r-f insula- faces when they are separated by an air gap or tion resistance of the crystal unit. Cl, Ll, and Rl other dielectric. If a dielectric exists on both sides represent, respectively, the distributed capaci- of the crystal, Ca would equal the total capacitance tance, inductance, and resistance of the electrical of the two capacitances in series. Thus, if the air- leads and terminals of the mounted crystal. Cl, in gap capacitances on the opposite sides of the crys- addition, includes the capacitance across any elec- tal were equal, as would normally be the case, Ca WADC TR 56-156 Figure 1-90, Equivalent circuit of crystal unit 70 would be equal to one-half the value of either one. C. is the electrostatic capacitance across the quartz plate, where the quartz serves as the dielectric. The series LCR branches represent the piezoelec- tric properties of the crystal as they appear to the external circuit when the crystal is undergoing mechanical vibrations. For this reason, these values are called the “motional-arm” (also, "series- arm”) parameters, in contradistinction to the parameters such as C« that are not of piezoelectric origin. 1-183. The motional-arm values of L are closely associated with the mass of the crystal, those of C are closely associated with the elasticity of the crystal, and the motional-arm values of R indicate the tendency of the crystal to dissipate heat during vibration. Each of the motional-arm branches is associated with a different mode or harmonic of vibration, and the normal frequency of each of the modes coincides with the series-resonant frequency of the respective LCR branch. It will be assumed that the branch indicated by L„ Ci, and R, repre- sents the equivalent circuit of the desired mode, and that all of the higher subscript branches Lk, Ck, Rk> represent unwanted modes. 1-184. Since a crystal unit is normally intended for use only within a very narrow frequency range centered at a specified nominal frequency, the equivalent circuit may be greatly simplified to that shown in figure 1-91. If the crystal is mounted so that the electrodes are in direct contact with the crystal faces, Ca will not be effective, and the values of L, C, and R in figure 1-91 will normally be approximately the same as those of L„ Ci, and R, in figure 1-90, and C„ will approximately equal Chi Chj Ce -|- Cl H . For these assumptions to Chj -|- Chj Section I Electrical Parameter* of Crystal Units hold, Ri must be much greater than the impedance of the crystal when parallel resonance is estab- lished between the motional arm and C«. Also, the operating frequency must not be so high that the reactance of Ll becomes significant ; and the nor- mal frequencies of all the unwanted modes must be sufficiently removed from the nominal frequency, if each of the unwanted branches is to present a high impedance at the desired operating frequency. SIMPLIFIED EQUIVALENT CIRCUIT OF AIR-GAP CRYSTAL UNIT 1-186. C, is normally much greater than the dis- tributed capacitance across the leads, so an air- gap or dielectric-sandwich type of crystal unit may be represented by the equivalent circuit shown in figure 1-92. This circuit, in turn, may be reduced to the equivalent circuit of figure 1-91 by assigning the following values to L, C, R, and C„ : C = Ca^C, (Ca + Ce) (Cl Ce Ca) CACe “ Ca -f Ce THE EFFECT OF R-F LEAKAGE RESISTANCE 1-186. The principal effect of Ri, the terminal-to- terminal r-f leakage resistance shunting the crys- Flgur* t-91. SImpfiRed »tfulvalant elreuH of crystal unit Figura 1-97. Slmplifiad equivalent circuit of air-gap crystal unit WADC TR 56-156 71 section I Electrical Parameter* of Cryatol Units tal, is to reduce the effective Q of the crystal unit. For all practical purposes this effect is negligible when the crystal is being operated at, or very near, the resonant frequency of the series arm. Under these conditions the electrical impedance of the crystal is so small by comparison that Ri can be ignored. On the other hand, as the frequency rises above the resonant point, the impedance in- creases sharply, and the greater the impedance be- comes, the greater is the effect of a given Ri. Insofar as the equivalent circuit of figure 1-91 is concerned, the effect will be to increase the value of R. The extent of this increase will depend upon how large the effective reactance of the crystal be- comes, relative to Ri. For the sake of simplification, most of the discussion given later concerning the equivalent circuit assumes that the increase in R due to Ri is negligible, or at least, is constant, re- gardless of the frequency, an assumption that can produce reasonably accurate results in the case of well-fabricated crystal units. The leakage resist- ance of military crystal units has a specified mini- mum d-c value of 500 megohms. As long as this minimum d-c value is maintained, R, at low fre- quencies will be comparable to this value. However, if an accumulation of moisture, dirt, or the like seriously reduces the d-c insulation resistance be- low the allowed minimum, the off-resonance char- acteristics will undergo a noticeable change. For instance, low-frequency filter crystals may have impedances at antiresonance in the neighborhood of 50 to 100 megohms. If Ri decreases below 500 megohms, the equivalent R of the motional arm will increase markedly. In the case of high-fre- quency crystal units, the effective dielectric losses may become relatively large, particularly when plastic holders are used, so that, at off-resonant frequencies, R, can become a significant parameter of the over-all effective resistance. However, for high-frequency crystal units employing modern methods of mounting and construction, R, can gen- erally be ignored. In the very-high-frequency range, crystal units are almost always operated at series resonance, so that, even if R, were on the order of 100,000 ohms, as might easily be the case, the effect would still be relatively minor. However, where the shunt resistance cannot be ignored, a more concrete analysis of its effect is to let R,, in figure 1-90, represent only the d-c leakage resist- ance, and to account for the r-f dielectric losses by inserting other equivalent resistances in series with the various shunt capacitances. In the sim- plified equivalent circuit in figure 1-91, the d-c leakage resistance could still be ignored, but non- negligible r-f shunt losses could be interpreted as WAOC TR 56-156 being due to a single resistance in series with the shunt capacitance, Co- The Q of the equivalent shunt arm will effectively equal the Q of the crystal unit when the unit is operated at frequencies well removed from resonance. The crystal units that are mounted in metal or glass holders of the type described in Section II, and are recommended for use in equipments of new design, can be expected to have shunt-arm Q’s greater than 1000 at all frequencies within their specified range. This as- surance, however, cannot be given for the crystal units mounted in plastic holders, particularly the old-style phenolic holder, or for those employing all-metal sandwich or air-gap electrodes. However, the lower Q’s of the older types of crystal holders are not entirely due to greater dielectric losses and larger values of shunt capacitance. An equally im- portant factor is the effective inductance of the circuit effectively in series with the shunt capaci- tances. For example, a corner-clamped air-gap mounting, such as that provided in a DC-31 crystal unit, has an effective shunt-arm Q of approxi- mately 80 or 180 at 30 me, depending upon whether the clamping pressure is applied by a coiled or a flat spring, respectively. Apparently, the reactance and resistance of a coil spring can be quite detri- mental to the quality of a crystal holder at very high frequencies, since it can cause not only an effective increase in the shunt capacitance, but also an increase in the effective dielectric losses. These losses would become prohibitive if the in- ductance of the spring and its stray capacitance should approach the properties of a series-resonant arm shunting the crystal. However, except in such abnormal cases, and in cases where the insulation is weakened by extremes in humidity and tempera- ture, the shunt resistance will have a negligible effect upon the performance of a crystal circuit. EFFECT OF DISTRIBUTED INDUCTANCE 1-187. The effective self-inductance of the crystal leads, Lu is normally not sufficient to seriously affect the crystal parameters, except in the case of very high operating frequencies where it is necessary to operate the crystal at series reson- ance. At resonance, the reactance of the crystal unit will be zero, so that the crystal, in combination with its shunt capacitance, must have a net equiva- lent series Xc equal in magnitude to the Xll of the distributed inductance. This means that the reson- ant frequency will be slightly lower than would be the case if there were no distributed inductance. The net effect on the equivalent circuit of figure 1-91 is that the LC product is increased very slightly (lower resonant frequency), and that C, 72 is increased to a greater extent. If the distributed inductance is completely negligible, the resonant frequency of the crystal will be slightly higher than the normal resonant frequency of the series arm, because of the reactive component of current through Co. However, the distributed Xl of the lower-frequency crystal units may be sufficient to approximately cancel the reactance due to the true Co. Under these conditions, the resonant frequency of the crystal unit as a whole would coincide with the natural vibration frequency of the crystal — an ideal operating state. In the case of the higher- frequency crystal units, the distributed inductive reactance may be sufficient to lower the frequency below the natural resonance point by several cycles. If the crystal unit were being operated at series resonance in a capacitance-bridge circuit, for ex- ample, such an effect would lead to frequency jumps with slight changes in the tuning adjust- ments. Under such conditions it would be desirable to add a capacitance in series with the crystal, with a reactance just sufficient to cancel the un- wanted Xll- The distributed inductance, Ll, of the lower-frequency crystals, and of practically any crystal unit which is to be operated above series resonance, has only a minor effect. The maximum effect will always be at very high frequencies near series resonance. In analyzing the behavior of a crystal unit where the distributed Xll cannot be neglected, the simplest approach is to consider Xi, as a separate fixed reactance in series with the crystal unit. From this point of view, as long as Xll is very small, as compared with Xo, it can be seen that Li, will not seriously affect the rate at which the net crystal reactance will change with frequency, and, therefore, will not influence the stabilizing effect of the crystal on the frequency. Crystal oscillators can operate successfully up to frequencies as high as 200 me. However, crystal- control of the frequency can be stable only when the impedance at series resonance is much smaller than the reactance of the effective shunt capaci- tance C„. The larger the value of Xu., the smaller this ratio will be. Thus, the higher the frequency, the greater the importance of keeping the crystal leads as short as possible, not only to reduce Ll, but also to reduce the distributed capacitance and the r-f resistance of the wires. The small coaxial- electrode type of mounting, such as the HC-IO/U, is the most satisfactory for achieving a minimum effective Co, and hence, a maximum frequency stability in the very-high-frequency range. It should be remembered, however, that since the dis- tributed Xll will increase with the frequency, the effective Co will also increase with the frequency. S*cKon I Electrical Parameters of Crystal Units SO that a measurement of C, at a frequency far lower than that of resonance will not alone give a reliable indication of the effective parameter near the operating frequency. EFFECT OF DISTRIBUTED CAPACITANCE 1-188. The effect of the distributed capacitance on the parameters of the simplified equivalent cir- cuit is merely to increase the value of Co. However, it should be noted that the amount of the increase will depend somewhat upon how the crystal unit is connected in the external circuit. For example, assume that the holder and terminal 1 in figure 1-90 are grounded. Chi, which would otherwise be in series with Chs, is now effectively short-cir- cuited, so that the total shunt capacitance C. is increased. If Chi were assumed to be equal to Chi, the amount of the increase due to grounding ter- minal 1 and the holder would equal Chi/2. On the other hand, grounding the holder can result in an effective decrease in C„. Assume, for instance, that a crystal unit is connected in a circuit equivalent to that shown in figure 1-93. With the metal holder ungrounded. Cm and Chi are effectively connected in series, so that, if Ch. = Chi, the total capaci- tance of the series combination is Ciii/2. If switch S is closed, thereby grounding the holder, the effec- tive total Co becomes larger or smaller, depending upon the point of view of the observer. Since Chi is no longer in series with Chi, but, instead, is shunted across the entire circuit, whereas Cm is shunted across the load Z, the total capacitance facing the generator is increased (assuming that Z is the reactance of a capacitor). When S is closed, the current through M, increases; how- ever, the current through Mj decreases. An ob- server at Ml would say that grounding the holder increased Co, whereas an observer at M, would say that Co has decreased. At frequencies well re- moved from the nearest resonant frequency of the motional arms, the branch impedances are so Flgura 1-93. Cirevit diagram Indicating the effect that grounding a motal holder may have on shunt capacitance WADC TR 56-156 73 Section I Electrical Parameters of Crystal Units high that the crystal unit behaves essentially as a capacitor of value C,. It can be seen that if a meas- urement were being made of the change in C„ due to the grounding of the holder, it would be impor- tant to know exactly how the measurements were made. For example, low-frequency crystal units mounted in the HC-13/U have been reported as having 0.8 ix/if less shunt capacitance, and medium- frequency crystal units mounted in the HC-6/U holder have been described as having 0.5 jn/if less shunt capacitance with the holder grounded. How- ever, it should be noted that crystal units so speci- fied are intended primarily for use in circuits where the crystal operates in a series-resonant rather than a parallel-resonant circuit. Even so, the grounded holder alters the entire circuit, not simply Co. Thus, in the circuit of figure 1-93, sup- pose that it is necessary for the current through Z to be in phase with the generator voltage. If the circuit is properly adjusted with an ungrounded holder, grounding the holder will detune the cir- cuit by effectively decreasing Co, on the one hand, and on the other, by shunting the load with Ch 2 . The over-all effect cannot be predicted simply by specifying an effective change in Co, since the end result will depend upon the impedance character- istics of the entire circuit. If the frequency of a crystal oscillator is being measured by beating its output with the output of a frequency standard, it is common practice to touch the crystal holder with the hand in order to determine whether the crystal unit which is touched is operating at a fre- quency above or below that of the standard oscil- lator. The oscillator frequency will be higher or lower than that of the standard according to whether the hand capacitance causes the beat fre- quency to fall or to rise, respectively, provided that the effective C„ is actually increased by the touch of the hand, as is invariably assumed. Before this assumption is made with complete assurance, however, the response of the circuit to a grounded holder should be known. EFFECT OF DISTRIBUTED RESISTANCE 1-189. Rl is assumed to include only the ohmic re- sistance of the electrical leads and the reflected resistance due to eddy currents in the holder and ground connections. At normal frequencies Rl is quite small, as compared with Ri; even the small- sized supporting wires of wire-mounted crystals have r-f resistances that are measurable in tenths of an ohm. As in the case of the other distributed parameters, the effect of Rl upon the equivalent circuit of figure 1-91 becomes more pronounced at the higher frequencies. To a first approximation R is simply R, -|- Rl. However, at frequencies above 10 me, the r-f resistance of the leads in- creases directly as the square root of the fre- quency, so that, in the v-h-f range Rl may be greater than one ohm. Rl will also increase some- what if the holder is grounded, as the increased eddy-current losses in the shielding will be re- flected as additional resistance losses in the crystal circuit. At normal frequencies, however, the effect of Rl is of minor importance; and even at fre- quencies above 100 me, its consideration is sec- ondary to the effects of the distributed capacitance. RULE-OF-THUMB EQUATIONS FOR ESTIMATING PARAMETERS 1-190. The crystal parameters for a given fre- quency vary rather widely from one crystal unit to the next. Even crystal units of similar dimen- sions and fabrication made by the same manufac- turer may show significant differences between corresponding parameters. These differences arise from the sensitivity of the quartz plate to slight changes in its angle of cut, surface state, elec- trode area, soldered connections, and the like. The parameter with the greatest percentage variation is R, and it is not uncommon for the larger values of R to be from 300 to 900 percent greater than the minimum values. The most predictable param- eter is C„ since it is primarily a linear function of the electrode area, the thickness of the quartz dielectric, and the dielectric constant, all of which are reasonably constant for a given fabrication technique, although variations may be expected in crystal units of the same nominal frequency and type of mounting, when made by different manu- facturers. For the same manufacturer, nominal frequency, and type of crystal unit, however, C, rarely varies by more than ±6% of its nominal value. With a reasonably constant as a starting point, approximate values for the major param- eters L, C, R, and C„ may be predicted for the principal types of crystal elements and holders. First, Ce is computed from the known values of plate area, dielectric thickness, and dielectric con- stant. Next, C can be found, since it is theoretically equal to C,. times a constant of proportionality. L can next be computed, since the LC product must conform to the nominal frequency. Next, an ap- proximate range of the values of R may be esti- mated from the empirical values of the crystal quality factor, Q. Since Q is the ratio Xl/R (or — Xc/R), R is thus equal to Xl/Q. Finally, C„ can be estimated by simply adding to C* the approxi- mate total distributed capacitance common to the particular type of holder and mounting. WADC TR 56-156 74 Estimating C„ Static Capacitance of Crystal 1-191. Althougrh the dielectric constant of quartz varies somewhat according to the angle of cut, the following formula will be approximately cor- rect for plated electrodes : C. = 0.402 A/t ixfit 1—191 (1) where A is the effective electrode area in square centimeters, and t is the thickness in centimeters. 1-192. In the case of partially plated A elements, where t is a function of the nominal frequency and the harmonic, equation 1-191(1) may be expressed as: Ce = 2.42 Af/n 1—192 (1) where f is the nominal frequency in mc/sec, and n, an odd integer, is the harmonic of the thickness- shear vibration. Although the quartz plates range from 1 to more than 2 sq cm in plate area, the electrode area normally covers only a fraction of the total quartz surface. The RTMA Standards Committee on Quartz Crystals has recommended the following approximate electrode areas for the fundamental frequencies of this type of crystal unit. Frequency in mc/sec (n = l) 1 — 2 2 — 5 5 — 9 9 — 15 15 — 20 Electrode Area ±10% (sq cm) 0,504 0.385 0.283 0.159 0.126 For the overtone modes, where n is greater than 1, the electrode area will be the same as that of the fundamental mode of frequency equal to f/n. The harmonics for various ranges of f are as follows : f = 10 — 45 me ; n = 3 f = 45 — 75 me ; n = 5 f = 75 — 105 me ; n = 7 1-193. In the case of crystals vibrating in a face- shear mode, it is the electrode area A that is a function of the frequency. For fully plated C ele- ments, equation 1-191(1) may be expressed as: Ce = 0.038/tf* pMf where t has an average value of 0.05 cm, and f (mc/sec) lies between 0.3 and 1 mc/sec. 1-194. For fully plated D elements, equation 1 — 191(1) may be expressed as: C, = 0.0172/tf^ Mpf SecKon I Electrical Parameters of Crystal Units 0 20 40 60 SO ’ 100 f (kc/sec ) Figure 1-94. C, versus frequency for typical wire- mounted N elements where t has an average value of 0.05 cm, and f (mc/sec) lies between 0.2 and 0.5 mc/sec. 1-195. For a typical wire-mounted J element, equation 1 — 191 ( 1 ) may be expressed as : Ce = k/f ppf where; k =: 38 for f = 1.2 to 2.5 kc/sec = 45 = 2.5 to 4.0 kc/sec = 58 = 4.0 to 6.6 kc/sec = 77 = 6.6 to 10.0 kc/sec Note that f in this case is to be expressed in kc/sec. 1-196. Typical values of C.. for an N element are shown in figure 1-94. Estimating C, Eguivalent Motionai-Arm Capacitance 1-197. After C, is known, an approximate value for C at the fundamental frequency can be readily obtained from the following equation : C = C„/re 1—197 (1) A ELEMENTS, l/t> 5 2 50 B ELEMENTS.l/t >5 650 C ELEMENTS.w/* = I 360 G ELEMENTS, o/C^.BS 35 0 J ELEMENTS,! /I <06 2 00 M ELEMENTS, «/e= .4 190 Figure 1-95. Approximate values of the ratio of capacitances, r^ = , for various plated crystal W elements WADC TR 56-156 75 Section I Electrical Parameters of Crystal Units where r, is simply the ratio of the electrostatic capacitance C, to the motional capacitance C, with C, and C expressed in the same units. The values of r^ for the more important elements are given :n figure 1-95. For the odd harmonics (n) of the thickness-shear modes: C = Ce/ren' 1—197 (2) Estimatin9 L, Equivalent Motional-Arm Inductance 1-198. Since Xl is equal to Xf at the series-reso- nant frequency of the motional arm, L is found quite simply, once f and c are known. Thus: L = 2^2 „ henries 1 — 198 (1) 4^ f C/ Remember, however, that f is expressed in cycles/ sec, and C in farads. Estimating R, Equivalent Met!onal-Arm Resistance 1-199. A theoretical equation for R would not be practical, since this parameter is much too sensi- tive to slight variations during the fabrication process and to changes in the crystal drive. An approximate estimate is gained from observations of the value of Q for the various frequency ranges. Thus: R = ohms 1—199 (1) Q where f is in cycles/sec, and L is in henries. The values of Q will range from 10,000 to 200,000, and in exceptional cases will have much higher values. Generally, the higher Q’s are to be found at the higher frequencies. For face-shear elements, the average Q is approximately 30,000, with most values falling between 10,000 and 40,000. Thick- ness-shear elements will have average Q’s of ap- proximately 75,000, and most of the values will lie between 35,000 and 100,000. 1-200. The Q is not a dependable parameter, and will vary from frequency to frequency, and from manufacturer to manufacturer, for the same type of crystal unit. For example, when' expressed as Q = — Xo/R = ^ it can be seen that Q is inversely proportional to C, and thus might be considerably increased by simply reducing the area of the electrodes. On the other hand, the re- sistance, R, is at least limited in practice by mili- tary specifications. For this reason, the typical 9000 8000 7000 6000 5000 4000 3000 2000 1000 900 800 700 600 500 400 300 200 lOO 90 80 70 60 50 40 30 20 I 0 c 7 - ( )K - “1 h V, L_ 3 n n \ 1 ■s r (n r:] '7 !ns 5) Kc »f XIO- L_ -n ± inrr: IT — — — Figure 7-96. Typical curves of the serres-arm resist- ance of plated crystals versus frequency. Actual series-arm resistances will vary between R/3 and 3R, where R is the value shown, except when R is less than 10 ohms, in which case the minimum resistance will be approximately one-half the value indicated. Values indicated are average for fundamental modes and approximately 'At the average for overtone modes values of R versus f, shown in figrure 1-96, are more likely to be found in randomly selected crys- tal units than is a given value of Q. The values of R indicated in figure 1-96 are merely typical, how- ever, and a small percentage of actual Military Standard crystal units will have series-arm resist- ances as small as one-third, or as large as three times the amounts showh. Estimofing Co* Totoi Static Shunt Copoeitonco 1-201. The equation for Co is Co = Co + Cd where Cd is the total distributed capacitance of the crystal leads and terminals. Approximate values of Cd for plated crystals in ungrounded holders are given below : Crystal Holder Cdiiifif) HC-6/U 0.7 HC-IO/U 0.3 HC-13/U 1.0 HC-15/U 1.5 WADC TR 56-156 76 IMPEDANCE CHARACTERISTICS VERSUS FRE9UENCY 1-202. The superiority of the quartz crystal as a frequency stabilizer lies in the fact that a small change in the frequency will cause a much larger change in the impedance of the equivalent circuit than can be obtained with conventional inductor- capacitor networks. Where an ordinary r-f tank coil would have an inductance measured in micro- henries, and an effective Q of 10 to 250, the equiva- lent circuit in figure 1-91 will have an inductance measured in henries and a Q of 10,000 to 250,000 or more. C, of course, is extremely small, since its reactance must equal Xi, at resonance, and is com- monly expressed in milIi-/*/*f (thousandths of a micromicrofarad). R is expressed in ohms, and although at low frequencies it may have values higher than 3000 ohms, depending upon the par- ticular crystal element and method of mounting, the more common values lie between 10 and 100 ohms. Co normally lies between 3.5 and 14 nnf, although much larger values are encountered where electrodes of large surface area are em- ployed. Among the smaller holders, such as types HC-6/U and HC-IO/U, values of 5 to 6 /i/xt are quite common. 1-203. Since Xi, = 2rrfL and Xc = -1 2irfC then, the rates at which Xl and Xo change with frequency will be, respectively : Section I Electrical Parameters of Crystal Units or or 2Tf.L = 1 2Tf.C 2irL 1 2s-f,*C However, note that this last equation not only implies that the two reactances have equal magni- tudes at the series-resonant frequency, f„ but also, that f, is the one frequency at which both react- ances will change with frequency at the same rate. Therefore, for small changes in frequency near series resonance: AXt, = AXc And since the total change in the reactance of the series arm is AX, = AX|, "1- AXc then AXg = 2 AXi. = 47rLAf If f. is taken as the reference frequency, so that af = f — f., then, since X, = 0 at resonance, the total reactance of the series arm, X„ will be equal to aX,. That is ; X. = 4TLAf 1—203 (1) Thus, for all frequencies near f„ the equivalent cir- cuit of a crystal unit may be represented as shown in figure 1-97, where Xc„ and R may be assumed dXc ^ 1 df 2xf"C Note that both of these derivatives indicate a posi- tive change in reactance with an increase in fre- quency. However, it should be remembered that Xc is negative, so that a positive change in Xo means that its magnitude becomes smaller as the frequency increases. On the other hand, the re- actance of the inductance increases by an amount 2irL for each additional cycle per second. At the series-resonant frequency of the series arm, the total reactance Xv+ Xc = 0 ’‘‘=o'"2,fsCo Xs * 4 v L Af Figure I -97. Impedance diagram of equivalani circuit of crystal unit 77 WAOC TR 56-156 section I Electrical Parameters of Crystal Units x« R. Xco[R* + X| + XeoX,] 3 m ' R' + [x,^):co]* I , LA» Xco RXio ^ R R' + [x, + Xco]* (3) = V^R* + Xj O @ Al = f-f. Figure 1-98. Cguiva/ent circuit of crystal unit when represented as an effective reactance in series with an effective resistance. The ganged arrows indicate that Xp and Rp will vary together with changes in the frequency, as indicated by the approximate formulas given as functions of Af. (Xco is negative.) to be constant, but with X, a variable that changes linearly with Af, and has the same sign as Af. 1-204. The series-parallel circuit of figure 1-97 may be reduced to an equivalent circuit of X, and Re in series, as shown in figure 1-98. It should be remembered, however, that X(,« is negative, where- as X. is either negative or positive, according to the sign of Af. The values of Re and X„ expressed as functions of Af, are not exact, but are close approximations, well within the accuracy of the normal test procedure, except when the numera- tors reduce to zero. Note, however, that with Af =r 0, the approximate expressions equate X, to 0, and Rp to R. This is equivalent to assuming that Xco is infinite by comparison with R, so that at series resonance of the motional arm the crys- tal unit as a whole behaves as a pure resistance equal to R. Although this is a close approximation, it is not exact. For X, actually to be zero, the term (R* -f- X,* -)- Xco X,) must be zero. There are two frequencies at which this will occur. One is called the resonant frequency of the crystal unit, fr, and the other is called the parallel-resonant, or antiresonant frequency, f,. RESONANT FREQUENCY OF CRYSTAL UNIT 1-205. First, it should be remembered that f„ the resonant frequency of the crystal unit, is almost, but not exactly, identical with f„ the series-reso- nant frequency of the motional arm. If there were no shunt capacitance, Co, then f, would indeed be the same as f, ; but, as it is, C„ introduces a reactive component to the current which must be cancelled by a reactive component of opposite phase through the motional arm, if the crystal unit is to appear as a pure resistance. These conditions are illustrated (not to scale) in the vector diagram of the currents through the two arms of the crystal unit, shown in figure 1-99. ‘Note that the frequency at which the crystal unit has the lowest impedance (maxi- mum current) is f.. Since X, = 0 at this frequency, the equivalent circuit of figure 1-98, according to * This sentence applies only to the relative impedances suggested by the current vectors in figure 1-99. It can be shown that the true minimum impedance of the crystal unit occurs at a frequency, f,„, that is as far below f, as f, is above f,. RE4CT(VEI total CURRENT AT SERIES + RESONANCE (fr) I( = total CURRENT AT SERIES RESONANCE OF MOTIONAL ARM (f, ) = Uo T I»s lcg= CURRENT THROUGH Co" CONSTANT 1., = CURRENT THROUGH MOTIONAL ARM AT f, 1., 'CURRENT THROUGH MOTIONAL ARM AT (, Iir«=-*Co ° reactive COMPONENT OF * resistive COMPONENT OF I,, I sri RESISTIVE I Figure 1-99. Fhasor representation (greatly exaggerated) of current through arms of crystal unit at the series-resonant frequencies, f, and f,. Distributed inductance of the crystal leads is assumed to be negligible WADC TR 56-156 78 equations 1 and 2, becomes and R. = r( R^ R" + Xc. Xr R"-|- Xc 0 ) Except at the very high frequencies, Xc„ is much larger than R, so that R, «= R, and X^ is so small that it may well be more than annulled by the dis- tributed inductance of the external wiring. Even at frequencies in the neighborhood of 100 me, Xq, will have a magnitude in the vicinity of 400 ohms, or approximately 10 times or more than that of R, so that Re will equal R within ±1 percent. The true frequency at which a “series-resonant” crystal circuit is intended to operate, however, is f,, where all the reactive components of crystal current cancel. Actually, the term “series-resonance” is somewhat misleading, for the conditions of crystal resonance are those of a parallel, and not a series circuit. It should be understood that when we speak of series-mode circuits and oscillators, the operating frequency is generally assumed to be f,. 1-206. By equation 1, figure 1-98, in order for X, to be zero, the frequency must be such that ; R* = - (X.^ -I- Xc„ XJ Since X(;„ is negative, this equality can only exist when X, is positive, i.e., X, is inductive, and f > f.. At frequencies very close to f., Xc„ > > X„ so that X,’ may be considered negligible. Thus, f^ will be the frequency at which R* = - Xc„X. = - 4 t LXcoAfr where Af, = f, — f. Since X^o is negative, Af = - R' ' 4tLXc„ 1—206 (1) Section I Electrical Parameters of Crystal Units According to figure 1-95, r, = 250. Thus, by equation 1 — 197 (1): By equation 1 — 198 (1); . ^ 10 '^ 4 X 3.14" X 10’" X 1.54 = 1.65 X 10 " henries According to paragraph 1-201: Co = 3.85 -1- 0.7 = 4.55 ixfil So that X = - 1 = - 10'" 27rf, Co 6.28 X 10" X 4.55 = - 3.5 X 10* n From figure 1-96, a typical value of R is found to be 8 n. On substitution of the foregoing values of R, L, and Xro in equation 1 — 206(1) , we find that: 4 X 3.14 X 1.65 X 3.5 = 0.088 cycle/sec. With such an extremely small difference between the two resonant frequencies of the crystal unit (less than 1 part of 10*), for all practical purposes it can be assumed that f, = f,. Indeed, it would be academic to seek to distinguish between them. Re- member, however, that the discussion has only concerned the equivalent circuit, in which the effects of the distributed inductance have been assumed to be reflected in a lower series-arm fre- quency,- and a larger C*. If the parameters in the example above are assumed to be the “true” values, so that the inductance of the leads must be represented separately, then a slightly more realistic interpretation will be possible. Assume, for instance, the Li, = ,10-' henries. Then 1-207. As a concrete example, assume that a par- tially plated A element, mounted in an HC-6/U holder according to RTMA recommendations, op- erates at resonance in its fundamental mode at a nominal frequency of 10 me. Approximately, what value of Af, could be expected ? Referring to para- graph 1-192, we find that A = 0.159 sq cm. On substitution in equation 1 — 192(1) ; C, = 2.42 X 0.159 X 10 = 3.85 X^L = 27rL,f, = 0.628 In order for Xe to cancel this reactance, then, by equation 1 in figure 1 — 98: - 4jrLAf, = 0.628 or Af, = - 0.628 4ir X 1.65 X 10 " = 3 cycles/sec WAOC TR 56-156 79 Saction I Elactrieal Parameters of Crystal Units The inductance of the external connections could easily increase this value of Af, ten-fold, so that for optimum frequency stability, an external series capacitance would be necessary. It is important to note the negligible effect that a small change in R or C« will have on the frequency of a crystal unit at series resonance. In equation 1 — 206(1), as applied to the 10-mc crystal unit, even if R should triple in value, the frequency would not change by more than 1 part in 10'. Although the power transferred through the crystal would be dimin- ished, as would the Q, and hence, the effectiveness of the crystal as a frequency stabilizer, a reason- able increase in R will not, in itself, cause the fre- quency of a series-resonant crystal oscillator to drift. ANTIRESONANT FRE9UENCY OF CRYSTAL UNIT 1-208. Returning again to equation 1 of figure 1-98, it can be seen that the term (R’ -j- X,* Xc® X,) can also be zero at some higher frequency than fr, namely, when X, = Xo, (R’ being negligible) . This would represent the high-impedance, parallel-reso- nant state of the equivalent circuit in figure 1-97. Letting af, = f, — f, then, at f„ the antiresonant frequency so that X„ = 4t LAf. = I Xco I IXcol 4ir 1—208 (1) On substitution of the typical values of Xco and L that were found for the 10-mc crystal unit: Af. 3.5 X 10^ 4 X 3.14 X 1.65 X 10-* 16.9 kc/sec For a 10-mc crystal, this value of f, represents a 0.169 percent frequency range in which the crystal may be used as a frequency-control device. At all frequencies within its range, except at f, and f., the unit will appear to the external circuit as an inductive reactance, X„ in series with a resistance, R,. There is a very simple relation between the Af fractional frequency range, -j*-, and the ratio of (2 the capacitances, r = ^, that can be derived from equation 1. Thus: "" 4,rL2irfCo so Af./f 1 2 (/ LCo where to = 2 t f. Now, 01 ^ = 1/LC so, on substitution: Af./f = C/2Co = 1/2 r 1—208 (2) In the case of plated crystals, r is usually some- what less than that predicted by theory. Where it should be slightly greater than the values of r, in figure 1-95, since C« > C., it is usually some- what less. However, as a practical rule-of-thumb, it can be assumed that r = r., but only in those cases where C. <=» C,. The ratio of capacitances, r, is quite an important parameter of the crystal unit in its own right, not only as an indication of the maximum percentage width of the frequency band in which a particular crystal element can operate, but, as will be discussed later, as a measure of the electromechanical coupling, and also, because of its relation to the frequency stability. IMPEDANCE CURVES OF CRYSTAL UNIT 1-209. Figure 1-100 shows the typical characteris- tics of the equivalent impedance circuit of figure 1-98, but with the frequency scale greatly ex- panded near the resonance point of the crystal. At frequencies sufficiently removed from resonance, both above and below f„ where the motional im- pedance is large compared with Xc., the X, curve is essentially the same as the reactance curve of a capacitance equal to C,. X, is inductive only be- tween its two zero points, f, and f,. Note that R, Figure 1-100. Impedance characteristic* versus fre- quency ef crystal unit. Neither the frequency nor the impedances are drawn to scale WAOC TR 56-156 80 rises sharply to a maximum at f., where it is equal to the parallel-resonant impedance of the equiva- lent circuit of figure 1-97. Since R is much smaller than Xcx,, at antiresonance Re = Z. = (Xc„)VR In the case of the particular 10-mc crystal unit where Xco = —3.5 X 10*n, and R = 8fl, Re at antiresonance will be approximately 1.5 megohms. Ze(= VRe* 4- ■^’) most frequencies is simply equal to Xe. Only in the immediate regions of fr and f„ where X* becomes negligible, is the magni- tude of Ze affected greatly by Re. The impedances, of course, are not drawn to scale. For example, if Z, at antiresonance were drawn to the scale used for Ze at resonance, the curve could extend more than a mile above the horizontal axis. PARALLEL-RESONANT FREQUENCY, fp, OF CRYSTAL CIRCUIT 1-210. Although an oscillator may depend upon a crystal operating at its series-resonant frequency, it is not practicable for a crystal unit to control an oscillator at the antiresonant frequency, f,. The crystal will either be operated to pass a maximum current (series-resonant circuit), or to develop a maximum voltage (parallel-resonant circuit) at some proper phase and frequency. It would seem that these latter conditions could best be met by operating the crystal unit at its antiresonant fre- quency, for it is in this region that the effective impedance is most sensitive to small changes in the frequency. However, another circuit, such as the input of a vacuum tube, will necessarily be Figure 1-101. Equivalent parallel-resonant circuit of crystal unit (X^p, R,p) shunted by load (X^., KJ. Nor- mally fr < fp < far M that X,p IS Inductlve and Xj. is capacitive Section I | Eloetrical Parameters of Crystal Units f shunted across the crystal. The shunt, or load cir- cuit into which the crystal operates will have a much lower impedance than that of the crystal at antiresonance, so that the total impedance will be relatively insensitive to small frequency varia- tions in the region of f.. In determining the actual frequency stability, the entire circuit must be con- sidered as a whole. The operating frequency may be considered as the resonant frequency, fp, of an equivalent parallel circuit, as shown in figure 1-101. Xe„ and R„„ are simply the reactance and resistance of the equivalent circuit of the crystal unit at fp, and X^ and R, are the equivalent shunt reactance and resistance, respectively. Since X, is more frequency-sensitive above series resonance than below, there is normally no advantage in using a crystal in circuits that require X, to be capacitive. Thus, in practice, f,„ will be some inter- mediate frequency between f, and f., so that X^p is always inductive and X, is always capacitive. The distinction made between “parallel resonance” and “antiresonance” in this discussion is somewhat arbitrary, and it is not uncommon to use the term “antiresonant” to describe any parallel-resonant crystal unit. Effects of Changes in Shunt Capacitance on fp 1-211. In discussing af,, the difference between the motional and the effective resonant frequency, it was found that That this quantity is normally insignificant is for- tunate, for it varies directly with the square of R, a parameter quite likely to change during opera- tion. On the other hand, it was later found that could amount to more than 0.1 percent difference in frequency. In this case, since, Af, is relatively large, it is also quite fortunate that, to a first approximation, the antiresonant frequency of a given crystal unit is independent of operational changes in R. However, it is not the antiresonant frequency of the crystal unit itself, but rather, the actual parallel-resonant frequency at which the crystal unit will operate that is of primary interest. Let fp — f, = Af„. Now, it can be imagined that Afp is simply the Af, of a crystal unit whose shunt capacitance C„ has been increased by an WADC TR 56-156 81 Section I Electrical Parameters of Crystal Units 2'Vt W. C,* Ct* » Xtp Ct* Cp+C, V S V 2 *»P *tp “ R+ Rj ’ R+Ry Figure I- 1 02. equivalent parallel-resonant tank cir- cuit, in which the motional impedance of the crystal unit is the inductive arm of the tank, and the total shunt impedance is the capacitive arm amount Ci and which has an effective resistance added to the shunt arm equal to Rj This last assumption can be made without introduc- ing an appreciable error as long as Rj is small compared with X,. The multiplying factor is needed, since only a fraction, f ^ ^ g total equivalent tank current will flow through R,. The equivalent tank circuit is shown in figure 1-102, where Ct and Rt are the values of the shunt parameters. Now, since Afp is equivalent to the Af, of a crystal unit that has C„ = Ct, Afp will be expressed by the same general formula that holds for Af,. Thus, Afp = . Also, since Xt =— l/2TfCT, it can be seen that Af,, will be inversely proportional to the total shunt capaci- tance. Although C„, itself, is the most stable of all the crystal parameters, the stability of the effec- tive external capacitance C, will depend upon the over-all design of the oscillator circuit. The crystal unit may be considered a device that determines the limits within which the frequency may be varied; that is, fp must lie somewhere between f, and f.. However, it is primarily the parameters of the external circuit in conjunction with the equiva- lent L, C, and Co of the crystal that fix the exact frequency ; and although the stability of the crys- tal parameters is fundamentally a problem for the crystal manufacturer, the stability of the effective Ct is largely the concern of the radio designer. 1-212. Figure 1-103 (A) shows the reactance curve of X, versus frequency, and figure 1-103 (B) shows the reactance curve of Xt versus Ct. The values of X, are those of the 10-mc crystal unit which has previously been taken as an example, and where L is assumed to be 1.65 X 10“’ henry. Note that the variations in Xt with frequency have been neglected, and f is simply assumed to equal the nominal frequency of 10 me, insofar as the capaci- tive arm is concerned. Since X, and Xt are drawn to the same scale, a horizontal line drawn through 01 23 4 S 67e9IOIIl2l3t4ISICr7IS 4 68l0l2l4l6l8 20 22 24 2«2e 30 32 34 3e3a40 Af (ke/KK) Crluut) WADC TR 56-156 Figure 1-103. Reactance curves of: (A) X, versus Af,(B) —X, versus Ct 82 both curvea will intersect points of equal but oppo- site reactances. These points of intersection will, in turn, indicate the value of af required for a given Ct, if the two arms are to be resonant. For example, at Ct = 7.69 /i/if, Xt = —2000 n ; so that in order for X, to be 2000 n, afp must equal 9.65 kc. Likewise, a Ct of 40 /i/»f will mean approximately a afp of 2 kc. Now it so happens that the part of Ct represented by C, will have a component that tends to vary with changes in the plate voltage applied to the vacuum tube, changes in the tem- perature or the tuning, changes in the coupling and neutralizing adjustments, and any changes in the vacuum-tube characteristics or other circuit parameters due to other causes. Such a change in Xi will cause not only a change in the resonant frequency, but also a change in the amplitude of the oscillations. If a given change in Ci is to have a minimum effect upon the frequency and power expenditure of the oscillator, then Cr must be as large as possible without seriously reducing the stabilizing effect of the crystal. In other words, Ct should have a value where the slope of the Xt-vs-Ct curve is not steep. For the 10-mc crystal of figure 1-103, maximum stability would be ob- tained with Ct between 36 and 40 /n^f. With C„ equal to 4.55 nid, this would mean a load capaci- tance, Ci, between 32 and 36 /i.#if. As much of Ci as is possible should be supplied by a fixed or ad- justable capacitor connected directly across the crystal unit or in some other part of the circuit, so that its effective capacitance with respect to the crystal terminals will remain constant, and not be affected by changes in the tube characteristics. This would reduce the variable part of C, to a minimum. Cr, however, should not be made so large that Xt will approach the magnitude of R, otherwise the crystal will not only lose some of its stabilizing effectiveness, but will require an excessive drive level to maintain oscillations. Sfabilbin<| Effect of Crystal on fp 1-213. Although Ct. plays an important role in the final determination of the frequency, it is the crys- tal itself that must be primarily responsible for the stability of the frequency — that is, if the use of a crystal is to be justified. For this reason, care should be taken to mdke certain that the apparent Q. of the crystal series arm as large as 10, if possible, and preferably much larger during operation. Otherwise, the series-arm impedance will not respond with maximum sensitivity to changes in Ct. However, since Xt, and hence X„ must be kept small to reduce the effects of a Section I Electrical Parameters of Crystal Units change in Ct, it might appear at first thought that a conventional coil could serve quite as well as a crystal. The reason why this is not true is that the frequency stability is dependent upon the mag- nitude of the change in reactance for a given change in frequency, and not primarily upon the total magnitude of the reactance. It will be re- called that, in the conventional L-C circuit, the in- stantaneous rate of change of Xl with frequency is and that, at resonance dXL ^ df df In the parallel-resonant crystal circuit, however, these equalities do not hold, for X„ = 4*rafL, and not 2>rfL. Thus, = 4irL, where L of the crystal is greater than L, of a coil of the same reactance by a factor of f/2sf. Since at resonance, the rate of change of Xt would equal that of Xi,, it follows that 4irL, the change in the motional reactance with frequency, will be fp/Afp times as great as the change in Xt with frequency. Consequently, the stabilizing effect of the crystal is much greater than that of the shunt reactance, so that, for ail practical purposes, the crystal can “automatically” annul the effect of small changes in Ct, but not vice versa. It can be seen that, even with X, rela- tively small, the stabilizing effect of the crystal for a fixed change in Xt is not diminished, pro- vided, of course, that X, is sufficiently large, as compared with R, so that the total impedance of the series arm is essentially equal to, and varies linearly with, X„. (See paragraphs 1-238 to 1-245.) Effect on Parallel Crystal Circuit Due to Variations in Resistance 1-214. As long as the apparent Q of the parallel- resonant circuit ^ ^ at least as great as 10, a change in either R or Rt will not, in itself, have a large effect upon f„. However, depending upon the design of the particular circuit, a change in the resistance may indirectly affect the fre- quency by causing a change in Cr, since, to a cer- tain extent, the effective Ct will be a function of the other circuit parameters. The most critical ef- fect due to changes in the resistance parameters is the effect on the power required for excitation of the oscillator in order to obtain a given output. The impedance, Zp, of the parallel circuit at reso- WADC TR 56-156 83 Section I Etectricol Parameters of Crystal Units X * nance will be approximately p .‘ ‘ *p . An increase Jv — p iVx in the total resistance of 100 percent would thus decrease Zp by one half. If , for example, the output of the oscillator depended directly upon the r-f voltage across Zp (i.e., across the crystal), a de- crease in Zp by one half would require twice as much power in the crystal circuit to maintain the output at the same level as before. A part of Rt will be the result of reflected resistance losses in the output circuit. An increase in the load will thus be reflected as an increase in Rt. This is unfortu- nate, for if the load should increase it would be desirable to have an increase in Zp, to raise the excitation voltage automatically, or at least to keep it constant. As it is, the effect is to decrease the excitation, unless special circuits, such as the Tri- Tet, are employed to increase the feedback directly. If a principal component of the losses in Rt are due to the losses in the grid circuit, and if the oscilla- tor design is such that the grid current is not linear with the excitation voltage, but rises at a much greater rate, then Rt can rapidly increase or decrease with the excitation voltage, and Zp will vary inversely. Under these conditions, Zp will always change in a direction that will tend to annul any change in the excitation voltage. The greater that part of Rt reflecting the grid losses, as com- pared with that part reflecting the output losses, the greater will be the amplitude stabilizing effect for counteracting changes in the plate voltage or the effective load resistances. Another character- istic of a crystal circuit in which Rt varies auto- matically is that the effect resulting from a varia- tion in R is minimized. Assume, for example, that the desired output at a constant load will require a certain effective value of Zp. If, for some reason, R should change, thereby changing the excitation voltage, Rt would tend to change by an equivalent amount in the opposite direction, thus maintain- ing Zp, and hence the output, essentially constant. However, a change in R or Rt will almost certainly be accompanied by a change in the crystal power losses, thereby causing a frequency drift if the particular crystal unit is frequency-sensitive to the drive level. At this point, however, the impor- tant items to note are: (1) Xt,, and hence X„ preferably should not be smaller than IO(R-I-Rt), or the maximum stabilizing effect of the crystal will not be realized; (2) the direct effect of a change in (R -j- Rt) is to change Zp; (3) the ef- fects of a change in Zp primarily will involve changes in the excitation voltage, in the power expended in the crystal circuit, as well as that de- livered to the load, and in the equivalent value of Ct, thereby also changing the frequency; (4) if changes in either R or Rt are such that the power expended in the crystal unit itself is caused to vary, then a significant change in the frequency characteristics of the crystal may result; and (5) for maximum frequency stability, the oscillator should be lightly loaded, and the drive level of the crystal should be as small as is practicable. Effect on Parallel Crystal Circuits Due to Variations in Motional-Arm C or L 1-215. Crystal circuits operated at the resonant frequency of the crystal units may be only slightly affected by variations in C or L from one crystal unit to the next, or even during the operation of a particular unit, provided that the effective LC product remains constant, so that the frequency does not change. In the parallel-resonant circuit, however, even if f, is the same, a different C and L means a change in Afp. For a given Ct and nominal frequency, Xtp, and hence, X.p, must remain ap- proximately constant, so that Afp, equal to X,p/4irL, will tend to vary inversely with L. The exact value of L for a given crystal unit will depend upon the effective electrode area, the orientation of the cut, the thickness of the crystal, whether twinning is present in the quartz, and the degree to which spurious modes are coupled to the desired mode. Insofar, as the variations in L from one crystal unit to the next are concerned, no problem arises unless it is necessary to adjust fp to an exact value ; in which case the problem of the design engineer is to ensure that Ct will be sufficiently adjustable so that the desired fp may be obtained with any reasonable value of L. Since such adjustments must be provided for anyway, in order to allow for different values of f„ no new problems are intro- duced, except that a greater deviation in Afp must be met than otherwise. Unless spurious modes are closely coupled to the desired mode, the variations in L that might oc^ur during the operation of a particular crystal unit will be too small to affect the magnitude of Afp, as long as Xtp remains con- stant. However, the operational variations of L and/or C may be such that f, will change, in which case fp will also change. Such a deviation in fre- quency, i.e., in the equivalent LC product, would occur during changes in temperature or drive level, or because of fatigue or other aging effects. Mini- mum variations in L and C are obtained by the use of low temperature-coefficient crystals and constant-temperature ovens, and by ensuring that the drive level will remain both low and constant. In any event, a reasonable operational variation in fp can be compensated for by an adjustment in Ct- WAOC TR 56-156 84 MialmHin Value of Afp 1-216. Returnin^r to equation 1 in figure 1-98, let it be imagined that X, represents the effective re- actance of the motional arm of a crystal unit in parallel with a total capacitance Cr, instead of simply the C« of the crystal unit, itself. Further- more, assume that Rt is negligible. As before, the condition of resonance is that X* be zero, which will occur only when R" -1- X/ -f- X. Xt = 0 (Note that Xt now replaces X,;„.) Now X, = 4irLAf, and on substitution in the preceding equation and rearranging, it is found that (Af)' Xt ^ =0 Note that this is simply a quadratic equation of the type AX’ -|- BX C = 0, so by the quadratic formula Xt /Xt' - 4R^ 4tL \ l&r'L' The ± term indicates that there are two possible solutions for Af at which resonance will occur. One of these is equivalent to Af, (but with Co replaced by Ct)» and the other is equivalent to Afp. For these solutions of Af to be real, Xt’ must be greater than 4R’; otherwise, the expression under the radical sign becomes negative, and Af will be imaginary. However, in the special case where Xt’ — 4R’ = 0, there is only one solution for Af. In other words. Af, Afp -Xt 8irL This represents the minimum value obtainable for Afp ; or, from the point of view of series resonance, it may be considered the maximum value obtain- able for Af,. The important point to note is that neither parallel nor series resonance is possible unless Xt* is equal to, or greater than, 4R’. At the minimum Afp, Xt* = 4R' or I Xt/R i = X./R = 2 It should be remembered that all the resistance has been assumed to be in the motional arm, and so Section I Electrical Parameters of Crystal Units has the effect of limiting the maximum component of lagging current for a given voltage ; but parallel resonance could be achieved at any frequency be- tween f. and f, if Rt were equal to R. However, since C„ limits the minimum amplitude of leading current, Rt cannot be made equal to R for all values of fp, and there will still be a minimum fp greater than f,. (A minimum which can be shown to be identical with the natural f, of the crystal unit.) With Rt assumed to be negligible, the ratio of reactance to resistance equal to 2 represents the minimum apparent Q. of the parallel crystal circuit, if resonance is to be obtained. As stated previously, if the full stabilizing properties of the crystal are to be in use, Q, should be at least 10. However, if the power delivered to the crystal cir- cuit is sufficient, oscillations can be maintained as long as the apparent Q. does not fall below 2. This occurs at the frequency at which the amplitude of the lagging component of current through the series arm is the maximum obtainable. TYPICAL OPERATING CHARACTERISTICS OF CRYSTAL UNIT 1-217. Figure 1-104 shows the effective impedance characteristics of the 10-mc crystal unit which has been assumed to have the following parameters: L = 1.65 X 10“* henry C = 1.54 X 10-’ /x/*f R = 8 ohms Co = 4.55 n/if Xe, R„ and are given by equations 1, 2, and 3, respectively, in figure 1-98; X(:„ is assumed to be equal to —3.5 x 10* ohms for all values of Af. Note that the normal operating range covers only about one fourth of the total range between f, and f,. Of course, if Co were greater than the value as- sumed, Af, would be smaller and the normal opera- ting range would be a larger percentage of the total. As explained previously, Ct ( = Co + C,) must be relatively large, so that small variations in C, will not greatly affect the frequency, and it is this consideration that limits the practical oper- ating range to low values of Af. At parallel reso- nance, Xe must approximately equal — X,, for the same reason that X, must equal — Xt. Standard military high-frequency crystal units are normally tested with a value of C, = 32 ii/if. At 10 me, a capacitance of 32 /i/,f will have a reactance of approximately — 500fJ, as indicated in figure 1-103 (B). With Co = 4.55 ^^f, Ct will be 36.55 ^id, which corresponds to a value of Xt = —440 O, and a Af = 2.1 kc/sec. Af can also be found from the reactance curve of figure 1-104 at the point where WAOC TR 56-156 85 Section I Electrical Parameter* of Crystal Unit* O I 2 5 4 S 7 e 9 10 II 12 13 14 IS 16 17 18 19 Af IN KC/SEC Figure 1-104. Typical characterittic curvet for X, and R, at 10-tttc crystal unit. (Shunt resistance, R/, across crystal is assumed to be negligible) Xe = — Xi = 500 fi. Crystal units are sometimes operated in series with an external capacitor, C,, as indicated in figure 1-105. Slight variations in the frequency can be compensated for by adjust- ments of Ci, and resonance will occur at the fre- quency at which X, is exactly annulled by X*. If the ratio of Xe/Re is sufficiently large, then for all practical purposes the series-resonant frequency, f„, is the same as the fp of the crystal unit in parallel with the same C,. At resonance, the crystal unit and Ci in series have an effective impedance equal to Re- Although there is an effective maxi- X Af mum Qpm = ^ when Af = , it has no special significance directly concerning the frequency sta- Figure 1-105. equivalent circuit of crystal unit connected in series with capacitor bility of the circuit, but does tend to increase the activity. In general, if a series capacitor is used, its reactance will be small, as compared with Xc«. Indeed, it may be used for no other purpose than to annul the self inductance of the crystal leads. It can be seen in figure 1-104 that R* does not in- crease nearly as rapidly as does Xe, except in the region of f,. With Ci = 32 /i/if, Af,, (= f„ — f,) will be 2.1 kc for the crystal unit of figure 1-104, and R, will be between 11 and 12 ohms. MEASUREMENT OF CRYSTAL PARAMETERS 1-218. The parameters L, C, R, and C„ of any crys- tal unit chosen at random are effectively four inde- pendent variables, so that a minimum of four measurements are required to determine the values of these variables. Probably the four easiest measurements to make are those for f„ R, Co, and f„. The measurement for the last quantity is made when a known load capacitance C, is connected in series with the crystal unit. Since fn, the resonant frequency of the crystal and Cj in series, for all practical purposes will be equal to the fp of the crystal in parallel with Ci, we shall normally not make a distinction between the two frequencies in the following discussion, but shall use the symbol “fp” in referring to either. WADC TR 56-156 «6 M*asiir«iii*at of tha Shunt Capucitaneu, C, 1-219. At all frequencies sufficiently removed from resonance the crystal unit will have the character- istics of a capacitance equal to C,. Thus, at these off-resonance frequencies, Co can be measured by a conventional Q meter or an r-f bridge. The fre- quency at which C. is to be measured should be lower than, but reasonably close to, the operating frequency, particularly if the crystal unit is to be operated at a high harmonic mode in the v-h-f range. Otherwise, the effect of the distributed in- ductance of the leads will not be properly taken into account. Muaturemeat of the Series-Arm Resistaace, R 1-220. R is normally measured with the aid of a Cl meter (crystal impedance meter). (See also paragraphs 2-60 through 2-6.5.) There are four standard Cl meters with which the crystal units described in Section II of this handbook have been tested : Crystal Impedance , Meter TS-710/TSM •TS-537/TSM TS-330/TSM TS-683/TSM Frequency Range (kc/sec) 10 to 1100 75 to 1100 1000 to 15,000 10,000 to 75,000 • (Crystal Impedance Meter TS-537/TSM may soon be replaced entirely by the recently developed Section i Electrical Parameter* of Cryctal Units Crystal Impedance Meter TS-710 ( )/TSM.) A Cl meter is essentially an r-f oscillator provided with a feedback circuit in which a crystal unit or a re- sistor, or a crystal unit in series with a calibrated capacitor, can be connected. A simplified schematic diagram of a typical Cl meter is shown in figure 1-106. The circuit shown is a modified Colpitts oscillator in which the tank inductor has been ef- fectively divided into two equal sections, L, and Lj, between which a resistor, R„, equal to R„ the effective resistance of the crystal unit, can be con- nected. The ganged tuning capacitors C, and Ci are at all times equal. Ch is simply a blocking capacitor to isolate the plate voltage from the crys- tal terminals, and adds only a negligible reactance to the tuned circuit. The potentiometer, P, is used to control the screen grid voltage, and hence the r-f output of the tube and the drive level of the crystal. With S, connected as shown, and R„ ad- justed to a value typical of the motional-rrm R for the type of crystal unit being measured, the circuit will oscillate at approximately the resonant fre- quency of the tank. If C, and Cj are adjusted so that the natural frequency of the oscillator is near the nominal frequency of the crystal, then, on con- necting the crystal into the circuit, oscillations will continue, but with the frequency determined WADC TR 56-156 87 Section I Electrical Parameters of Crystal Units by the total reactance of the tank, including X, of the crystal unit. There is no standard practice as to grounding the crystal holder; but whether grounded or ungrounded, the method of connect- ing the crystal unit should be noted. The drive is adjusted so that a very small grid current is indi- cated on M,. Under these conditions the control grid is positive with respect to the cathode only at the peaks of the positive swings of the excita- tion voltage developed across C,. Since electrons flow from cathode to grid only at these instants, a small percentage change in the excitation volt- age, as illustrated in figure 1-107 can cause a very large percentage change in the cathode-to-grid electron flow. In an actual circuit the idealized constant bias that is indicated in figure 1-107 does not occur because of the gridleak action. However, if the gridleak contribution to the bias is very small compared with that part developed across the cathode resistance, the increase in bias due to an increase in excitation occurs almost entirely across R*, not across the cathode resistance. Hence, a five or ten per cent increase in the total bias can result from hundred per cent increase in the gridleak IR drop. In this way, the grid current meter is a very sensitive indicator of slight changes in the r-f voltage across C„ and hence of any change in the tank current. With S, in the crystal position and S 2 closed, as C, and C, are varied, a peak in the grid-current reading indicates a maximum current through C,. This in turn means that the effective resistance of the crystal unit has reached the minimum value equal to the series-arm R. In other words, the oscillator fre- quency is coinciding with the resonant frequency, fr, of the crystal unit. R„ can now replace the crys- tal in the circuit and be adjusted to give the same meter readings at the same frequency. At this point Ro will equal R, and, since R„ is known, R will have been measured. The crystal current meter, M 2 , is not sufficiently sensitive to permit an accurate observation of the small changes in tank current that occur as the circuit is tuned through f,. The purpose of the meter is to decrease the possibility of overloading the crystal and to pro- vide a ready me&ns for determining the exact drive level at which the crystal is being tested. Since the crystal parameters may change with the drive, it is necessary to specify the drive level at which the measurements are made. Expressed in milliwatts, *the drive level equals PR X 10~S where I (in milliamperes) is the current through Mj at f,. If a crystal-current meter is not supplied, two vac- uum-tube voltmeters can be used to measure the voltage from each crystal terminal to ground with- out seriously affecting the circuit. Where E is the difference in potential across the crystal, equal to the difference between the two terminal voltages, the drive level is equal to EVR- The temperature at which the measurements are made should also o UJ a. CO o o OEO o X -J < o o f o Z I- o o o lU o < o > LJ q: a: D o a: o Figure 7-707. How a small percenfage change in excitation voltage can cause a large percentage change in grid-leak current of Cl meter. Bias does not actually remain constant as indicatedf but follows the percentage changes of excifafion voltage. Nevertheless, the relative percentage variations of grid current and excitation voltage can be approximately as shown when the greater part of the bias is developed across the cathode resistance WADC TR 56-156 86 be specified, although, in general, the variation of R with ambient temperature is much less than its variation with the amplitude of the crystal vibrations. Measurement of the Resonance Frequency, f, 1-221. To measure f,, a c-w radio receiver, a radio- frequency standard, a calibrated audio-frequency source (interpolation oscillator), and either a loudspeaker, a pair of ear phones, or an oscillo- scope are used in conjunction with the Cl meter. With the crystal connected in the Cl-meter circuit and the oscillator tuned to series resonance, the r-f output can be loosely coupled through a coaxial cable to the antenna post of the c-w radio receiver. After the receiver is tuned to the frequency of the crystal unit, the Cl meter is turned off, and the receiver is connected and tuned to receive the par- ticular harmonic of the frequency standard that is nearest to f,. The bfo of the receiver is then cut off, and the Cl meter is turned on. With both the standard and the Cl-meter signals being fed to the receiver input, the output of the receiver will be an audio beat note equal to the difference be- tween the known standard frequency and the un- known crystal frequency. By momentarily switch- ing a fairly large value of C, in series with the crystal, so that the Cl-meter frequency increases slightly, the audio beat frequency will rise or fall according to whether f, is respectively greater than or less than the standard signal. The audio beat frequency is next mixed with the audio out- put of the interpolation oscillator, which in turn is adjusted to bring the beat frequency of the two audio signals to zero — the zero beat being ob- served by phones, loudspeaker, or oscilloscope. At zero beat, the crystal frequency will have been measured to be equal to the selected r-f standard frequency ± the interpolation oscillator frequency. The accuracy of the measurement depends pri- marily upon the accuracy of the frequency stand- ard, and secondarily on that of the interpolation oscillator. As in the case of the resistance meas- urement, both the temperature and the drive level should be specified, and these should be the same as when the measurement of R was made. Measurement of the Parallel-Resonance Frequency, f,, 1-222. To measure f,„ it is first necessary to adjust the Cl meter to oscillate at f„ by the same steps employed previously. With the oscillator so ad- justed, a known value of is switched in series with the crystal. The new frequency will be ap- proximately equal to fp. The more common values Section I Electrical Parameters of Crystal Units of Cl are 32 /i./if for high-frequency crystals, and 20 ([t/if for low-frequency crystals. In testing to determine whether the crystal frequency is above or below the frequency of the test standard, it may be more convenient to add capacitance across the crystal unit than to change the setting of C^. In this event the Cl-meter frequency is decreased rather than increased, so that the effect on the beat note will be the opposite of that previously described. Simply touching the crystal holder with the hand is normally the quickest method of in- creasing the shunt capacitance; however, care should be taken that the method employed does not effectively decrease the capacitance by ground- ing the holder. 1-223. Theoretically, the foregoing method of measuring f„ is not exact, for if the LC circuit is correctly tuned when the crystal appears as a pure resistance, the same feed-back phase relations can- not hold at the higher frequency, f,„ unless CR and C, in series introduce a negative reactance to compensate for the increase in Xj,, and Xlj and the decrease in Xc, and X^j. In other words, X. of the crystal unit is approximately, but not exactly, equal in magmitude to X, of the load capacitance. Actually, Xe is less than X,| by an amount ap- proximately equal to the change in reactance around the LC loop, exclusive of CR and C,. This change is approximately equal to 4jr(L, -|- L,) Afp, where Af,, is the difference between f„ and f,. At the true fn, X, -f X, = 0; at the observed fp, Xe -h X, -H 4ir (Li L 2 ) Afp = 0 1—223 (1) Now a small change in X,. -j- X^, equal to aX* -4- aXx, as a result of a small change in frequency, is practically equal to AXe alone. If the frequency is sufficiently close to the resonant point, f„ we may set X, -f X^ (at observed f,,) = A(Xe -f- X,) == aX, <== aX, = 4-rLAf, where Af = observed fp — true fri. By substitution in equation (1) True == Observed fp + ^ ^ ^ ^ Afp 1—223 (2) Maasaramanf of the Effective Resistance, R„ at Parallel Resonance 1-224. In the measurement of fp, the drive level and the temperature should be the same as in the measurement of R and f,. To determine the drive level, either the voltage across the crystal unit, or WADC TR 56-156 89 Section I Electrical Parameters of Crystal Units the effective resistance should be known — or both, if a crystal-current meter is not provided. A measurement of Re is also important for its own sake and as a check to see whether the motional- arm parameters are the same at fp as at f,. In the case of a crystal unit which is intended to be oper- ated only at parallel resonance, R, is generally treated as a primary parameter of more immediate importance than the motional-arm R. R, is meas- ured in a manner similar to the measurement of R, except that on substituting R„ for the crystal the circuit must be retuned so that oscillations are being maintained at fp. For a very precise drive- level measurement, additional precautions must be taken if the power dissipation is to be the same ■ n both the series- and parallel-resonant measure- ments. The best assurance that the f, and fp drive levels will not be greatly different is to be had when the crystal current is kept near the minimum necessary to maintain oscillations. Thus, even though the relative differences in drive level may be large, the absolute differences will be small. This is not a completely reliable method, for some crys- tal units exhibit very sharp increases in resistance when the drive level approaches a minimum. Computing the Series-Arm C and L from the Measured Parameters 1-225. From the formulas for f,, Xp, and Xi, it is quite easy to derive the following approximate equations for the series-arm parameters, C and L : C = 2(Co -f CJAfp fr (27rf,)^C 1—225 (1) 1—225 (2) where, C, C„, and Cx are in farads, L is in henries, and Afp and f, are in cps. METHODS FOR EXPRESSING THE RELATIVE PERFORMANCE CHARACTERISTICS OF A CRYSTAL UNIT 1-226. If the four equivalent electrical parameters (L, C, R, C„) are accurately known for a given state of operation, no other independent data con- cerning a crystal unit can increase the radio engi- neer’s knowledge of how the crystal will perform under the given conditions. However, the radio en- gineer has been slow in requesting specific infor- mation concerning the electrical characteristics of the crystal units available, and as a result the prob- lem of making a given circuit perform correctly has often in the past effectively become the re- sponsibility of the crystal manufacturer, who, by cut-and-try methods, has been more or less required to design the crystal unit around the particular circuit. Fortunately, progress toward greater standardization of crystal units has been considerably accelerated during recent years be- cause of the increased demands of the military services ; but there is still a tendency on the part of the design engineer to regard a crystal unit, as one production engineer has expressed it, as a “mystery box," rather than the equivalent circuit that it is. Contributing to this tendency has been a hesitancy upon the part of the manufacturer to describe his crystal units in terms of the most probable equivalent electrical parameters. At the present state of the art, wide variations from the most probable values can occur, and the manufac- turer quite naturally wishes to avoid the chance that typical values of the parameters will be mis- interpreted as specified values. For similar rea- sons, a description of a crystal unit in terms of its most probable parameters is not at present desira- ble from the point of view of the military services, lest a crystal circuit be designed upon the assump- tion that the typical crystal parameters will always be available, rather than upon the assumption that the crystal unit cannot be depended upon to meet other than its minimum performance specifica- tions. If the former, rather than the latter assump- tion were made, a carefully designed circuit might fail to operate properly if used with a borderline crystal unit. Thus, the purpose of the standardiza- tion of types — to ensure a complete interchange- ability among the crystal units of the same type number and nominal frequency — would be de- feated. Nevertheless, the lack of emphasis upon the basic parameters has served to cloak the crys- tal in an air of mystery, and to instill in the radio engineer an impression that a crystal circuit is possessed of properties that cannot be expressed in the normal idiom of LCR networks. Contribut- ing somewhat to this point of view is the special terminology that has been developed for the pur- pose of comparing the performance characteris- tics of one crystal unit with those of another par- ticularly where the definitions of the terms contain certain ambiguities or conditional interpretations, or are presented as mathematical relationships without concrete qualitative meanings. What may be implied as a property of the crystal unit alone, may well be a function of the particular circuit in which the crystal unit is mounted. Much of the difficulty can be avoided if it is kept in mind that a crystal unit has no important circuit performance WADC TR 56-156 90 qualities that cannot be expressed in the everyday terminology of radio engineering as it might apply if the crystal unit were replaced by an equivalent network of L, C, R, and C». 1-227. There are five general categories in which crystal units can be placed for comparison insofar as their relative merits are reflected by their per- formance in a standard test-oscillator circuit : ( 1 ) activity, (2) frequency stabilization, (3) band- width, (4) quality factor, and (5) parameter sta- bility. Activity, as applied to a crystal, is a general term, rather loosely defined, that refers to the relative ease with which a crystal may be caused to maintain oscillations. The basic parameter most closely associated with the crystal activity is the motional-arm resistance, R. Besides R, or R^, there are certain performance parameters that can be used as indices of relative activity quality. These are the effective Q (Q^), the maximum effective Q (Qcm), the figure of merit (M) , and the perform- ance index (PI). The term, frequency stabiliza- tion, as used in this context, refers only to the ability of a crystal to minimize any change in the frequency due to variations in the parameters of the external circuit, In this sense, those perform- ance parameters that can be used as indices of the frequency-stabilization quality are the series-arm L/C ratio, the coefficient of frequency stability (Fx), and the capacitance ratios Ct/C and Ct/Cx- The bandwidth of a crystal unit refers to the fre- quency range over which the crystal unit is con- sidered operable. The performance parameters indicating this quality are the capacitance ratio, r = Co/C, and the electromechanical coupling fac- tor, k. The quality factor is simply the crystal Q, which is, itself, a major performance parameter, but one that is not exclusively identified with any one of the other four performance categories. The term parameter stability is used here to refer to the relative stability of the crystal parameters during changes in the temperature, drive level, tuning adjustments, and the like. The frequency stability of the crystal unit, which is included in this category, should not be confused with the function of frequency stabilization which is the characteristic we have arbitrarily assigned to the second performance category. The frequency sta- bilization is dependent upon the magnitudes of the equivalent-circuit parameters; whereas, the fre- quency stability is dependent upon the stability of the equivalent-circuit parameters. The stability of a crystal oscillator circuit is dependent upon both the crystal stabilization and the parameter stabil- ity. Performance indices or terms indicating the relative parameter stabilities are represented by Section I Electrical Parameter* of Crystal Unit* the temperature coefficients of frequency and re- sistance, drive-level coefficients of frequency and resistance, frequency tolerance, frequency devia- tion, resistance deviation, relative freedom from unwanted modes, and general expressions indi- cating durability and aging characteristics. Since most of the characteristics identified with the five performance categories can be expressed as func- tions of the same basic equivalent-circuit param- eters, a performance parameter in one category quite often serves as an indication of the crystal quality in another. It cannot be said that those properties most closely identified with the activ- ity, foe instance, are not also related to the fre- quency stabilizing effect. Nevertheless, classifying the various methods for rating the performance of a crystal unit is helpful in interpreting the dif- ferent performance parameters in terms of the basic equivalent-circuit parameters. Activity Quality of Crystal Unit 1-228. The “activity” of a crystal oscillator is a qualitative expression referring to the amplitude of the oscillations. It is a term that came into use during the early days of crystal resonators, but one that seems never to have been vigorously de- fined. For example, it is not always certain whether the “activity of an oscillator” is intended to refer to the amplitude of current in the feed- back, or in the output circuit, or to the voltage across some particular circuit component, or to the output power, or to the excitation power, or to the ratio of these powers, or simply to the ampli- tude of the crystal’s mechanical vibrations. Were the expression not already so strongly entrenched in the crystal terminology, its use would probably be discouraged. As it is, crystal units are com- monly described as having high or low activities, or more specifically, as having high or low poten- tial activities or activity qualities. It will be found that the crystal parameter most directly indica- tive of the activity quality is the motional-arm con- ductance, 1/R. In crystal oscillators employing gridleak bias, when one crystal is replaced by another of the same nominal frequency, one of the crystals is usually found to produce stronger ex- citations and hence a larger grid current under similar operating conditions. Frequently the rela- tive grid currents are defined to be equal to the relative activities of the crystals. With this method of measurement it can be seen that, if a crystal is connected directly across the grid-to-cathode input the excitation, and hence the activity, will depend upon the amplitude of the r-f voltage across the crystal. On the other hand, if the crystal is con- WADC TR 56-156 91 Section I Electrical Parameters of Crystal Units nected in series with the oscillator input, the activ- ity will depend upon the amplitude of the current through the crystal unit. Since, in any event, the grid current depends upon the values of every parameter in the oscillator circuit, such a meas- urement is ambiguous unless a standard test cir- cuit can be referred to for each frequency. Only in this way can the crystal unit, itself, be con- sidered the only significant variable. Even under the assumption of ideal test conditions, however, the exact mathematical relationship among the crystal parameters, which provides the most direct measure of a crystal unit’s inherent activity qual- ity, has been a subject of some controversy. A number of suggestions have been made, but the usefulness of each of these depends considerably upon the method by which the crystal is to be used to control oscillations. As the crystal terminology becomes more rigorously defined we can imagine that the word “activity” will fall into disuse even- tually, with “effective resonance resistance” or “conductance” taking its place. ACTIVITY QUALITY FOR SERIES RESONANCE 1-229. As an example of series-mode operation, we refer to the test circuit in figure 1-106. It can be seen that the grid excitation will be approxi- mately equal to IXc, the r-f voltage developed across C,. X<:, depends upon the frequency and the value of C„ whereas, I, the current through Ci depends upon the B+ voltage, the setting of P, the tube characteristics, etc., as v.'ell as the tank im- pedance and hence the resistance of the crystal. With all the circuit parameters constant, the only variable that the crystal introduces at resonance is its resistance R. Rather than specify all the parameters of the test circuit for each nominal frequency, it is clear that the measurements of R provide sufficient indication of the relative activi- ties of different crystal units under any similar conditions of resonance. Since the current and voltage amplitudes vary inversely with R, the series-resonance activity of any crystal unit can be assumed to be directly proportional to 1/R, the motional-arm conductance. ACTIVITY QUALITY FOR PARALLEL RESONANCE 1-230. The interpretation of the activity quality of a crystal unit becomes more complicated when the crystal is to be operated at parallel resonance. But even as in the case of series resonance, the inher- ent pi'operty of the crystal unit that most readily indicates the relative activity is the motional-arm conductance, 1/R. In a parallel-resonant oscillator circuit, the excitation is normally directly propor- tional to the voltage developed across the crystal in parallel with its effective load capacitance Ci. This voltage in turn is proportional to Zp XtVR. where Zp is the parallel-resonance impedance, Xt is the reactance of Ct, the total shunt capacitance, and R is the series-arm resistance. As long as Xt remains constant, the only significant crystal vari- able that affects the activity is R, or more directly 1/R. A complication arises from the fact that Ct = Co -|- Ci, so that if Ci, the effective capacity of the external circuit is to be held constant, then Xt, and hence the activity, changes with C„. Another complication arises when a measure of crystal quality is desired that will hold between crystals of different nominal frequency. If Ct or Ci is to be held constant, then Zp tends to vary inversely with the square of the frequency. Fi- nally, the complications are multiplied several fold when one begins to take into account the many ways in which a crystal can be connected to sta- bilize or to control a parallel-resonant type of cir- cuit. Unless the term “activity” is to refer to some desired and well-defined end result that can be measured quantitatively, the word may have little practical meaning. With this in mind, we note that since the usefulness of the oscillator depends en- tirely upon its output, the useful oscillator activity can be said to concern only the amplitude of oscil- lations in the load circuit. Thus, the relative ease with which a crystal unit enables a given output to be achieved under specified conditions can be re- garded as the relative activity quality of the crys- tal unit. From this point of view, the relative activity of a crystal unit can be considered in- versely proportional to the driving power that the crystal requires in order to maintain a given out- put level in a fixed load. Now, there is a special case of parallel-mode operation — where the series arm operates into a constant Ct — for which the activity, as interpreted above, requires only a measurement of R. Assume that a small, variable shunt capacitance, Cy, can be connected directly across each crystal unit whose activity quality is being tested, so that the effective shunt capaci- tance (eff Co = Co -h Cy) can be adjusted to give the same value for all crystals. With this arrange- ment, the crystal power dissipation for a given power output will vary positively with the mo- tional-arm resistance, R. In other words, two crys- tal units of the same series-resonance frequency and the same motional-arm resistance can produce the same oscillator activity at the same parallel- resonance frequency regardless of the particular WADC TR 56-156 92 / 1 values of Co, or of the motional-arm C and L, but only if a variable shunt capacitor is provided by which the effective Co can be held constant. Thus, for gauging the potential activity of a particular crystal unit to be used in any parallel-mode oscil- lator circuit having a variable capacitor connected directly across the terminals of the crystal unit, the motional-arm conductance, 1/R, can usually be considered a . sufficient and proper activity parameter. 1-231. Where the load capacitance, Ci, that the entire crystal unit faces (not necessarily Ct = Co -|- Cl that the series arm faces) is to remain constant, the parameter 1/R is not a sufficient index of the activity quality. If the proper meas- ure of crystal activity is defined to be the ratio of output power to crystal power, such a definition is general enough to be applicable for any type of crystal oscillator. Certainly, the crystal unit that requires the least expenditure of energy to per- form its task should be considered the one of great- est activity. Unfortunately, activity quality, from the point of view of a power ratio, becomes a function of each particular oscillator and load cir- cuit, so that the generalization gained in the defi- nition is completely lost on application, unless a standard test oscillator is available for each type of circuit. Only when the crystal unit is operated at its series-resonant frequency or is connected directly in parallel with a variable capacitor, can the relative activity qualities of two or more crys- tal units be considered constant and independent of the particular design of the external circuit. In all other cases, Co, as well as R, becomes a signifi- cant parameter of the activity, and the exact rela- tion of C« to the activity will depend upon the circuit design. For a parallel-mode activity param- eter to apply in the general case, the oscillator circuit, itself, must be considered from a general- ized point of view. By this approach, the effective Q I^Q, =^^is often considered a more reliable activity quality factor, than 1/R alone. The reason for this belief is most readily indicated when a generalized crystal oscillator is represented by the negative-resistance method — in particular, by diagrams (A), (B), and (C) in figure 1-108. Q, AS AN INDEX OF ACTIVITY QUALITY 1-232. If the oscillations of a crystal are to be sus- tained, energy must be supplied at a rate equal to the power losses in the crystal. This state is indi- cated in figure 1-108 (A), where the power source is represented as a generator with an emf equal. Section ^1 Electrical Parameters of Crystal Units but opposite in sign, to the voltage across the effec- tive resistance of the crystal unit. The power input thus is |EIo| =:U*Re. Furthermore, since the total voltage drop around the circuit must be zero, the external circuit must appear as having a reac- tance equal, but opposite in sign, to the effective reactance of the crystal. Now, since the current through each component of the equivalent series circuit is the same, the voltage may be represented as being the result of a current flowing through a circuit of zero total impedance, as shown in fig- ure 1-108 (B) . Note that the generator is replaced by a negative resistance, p„, numerically equal to R,. Figure 1-108 (C) shows the same operating conditions, but with X, and of the external circuit replaced by an equivalent capacitive reactance, X„ in parallel with a negative resistance, p, equal X ’ X * to — Z,„ where Zp = ^ At all instants Jt«f> the impedance across the terminals at 1 and 2, whether that of the crystal unit or of the external circuit, is the same for both the (B) and the (C) equivalent circuits. Imagine now that after equi- librium has been reached, R,. suddenly decreases by one half. At this instant the power being sup- plied, UV. = Up, is greater than that being dissi- pated in R,. In (B), the amplitude of oscillations will increase until a new equilibrium is reached, at which time p. will also have decreased by one half and will be once again numerically equal to R,. In (C), the halving of R^ means that Z,, is approximately doubled. The same increase in cur- rent through Re must be shown to occur in (C) as in (B). Thus, the amplitude of the oscillations must increase until p has doubled its value and is again numerically equal to Z,,. (The changes in p are caused by the limiting elements in the oscilla- tor. For example, R,, of the vacuum tube will in- crease with an increase in gridleak bias, and this w'ill be reflected as an increase in p.) From the generalized circuit approach, we can reach the general conclusion intuitively that oscillations build up as long as |p,i > R„ or |p < Z,,, and that oscillations diminish in amplitude under the re- verse conditions. It may hot be at once apparent why oscillations should not build up if p were nu- merically larger than Z,„ in the same way that they do when p, is greater numerically than R,. A rigorous proof can be obtained by a differential equation of the current through the inductance, applying Kirchoff’s laws and keeping in mind that resistance, negative or positive, is mathematically an instantaneous rate of change of voltage with current. Qualitatively it can be seen that the amp- litude increases or decreases, depending upon WAOC TR 56-156 93 Section I Eiectrieai Parameters of Crystal Units Figure J-IOS. Generalized crystal oscillator circuit whether the ratio of the power input to the power dissipation is, respectively, greater or less than 1. For circuit (B) the current, Ic, is the same for both the crystal unit and pt, so that the power ratio is : 1/ p. Ps I.*R. Re 1—232 ( 1 ) Oscillations thus build up as long as p, is greater than R,. For circuit (C) the voltage, E„ is the same across the crystal unit as across p, so that the power ratio is: EqVp ^ Zp R. EoVZ.* p where 2^. = VRe* + X,* and Xe > > R, 1—232 ( 2 ) Oscillations thus build up as long as p is less than Zp, the equivalent parallel-resonance impedance at equilibrium. The initial values of p, and p for a given frequency can be assumed to be fixed param- eters characteristic of the particular oscillator circuit, although the exact magnitudes may be ex- tremely complicated functions involving all the circuit variables. Nevertheless, it is reasonable to assume that the more the negative resistance must change, the greater will be the activity of the os- cillator by the time equilibrium is reached. From the point of view of circuit (B), it would seem that with a given initial p, maximum activity is to be obtained with a minimum Rei but in circuit (C), on starting with a given value of p, maximum activity is to be obtained with a maximum Z„. Note that these two conditions are not entirely equiva- lent. For a given crystal unit, Zp, for instance, can be increased by increasing X„ which in turn re- quires that Re, as well as X,, become greater. (Since X, must increase to match the increase in Xi, so must the frequency, and hence also R*.) Remembering that the activity that is assumed to WADC TR 56-156 94 be proportional to the change in negative resist- ance is that in the oscillator output and is not necessarily the current amplitude in the crystal circuit, it can be seen that R» alone, in spite of the implications to be drawn from figure 1-108 (B), may not be a sufficient parameter to indicate the relative activity quality of a crystal unit in the general case, i.e., X, must also be considered. For these reasons, the effective Q of the crystal unit, Qe = Xe/R„ is usually considered the more reliable index of the crystal activity quality for parallel- resonant oscillators. There are exceptions, how- ever, where Re, or rather is the proper activity parameter. These occur when the crystal is actu- all.v operated at series resonance with an external capacitance. An example is the Cl-meter circuit in figure 1-106, when Ci is connected in series with the crystal. (See paragraph 1-585 for a more de- tailed analysis of negative-resistance limiting.) 1-233. The crystal Q, is a more direct index of the potential activity in some oscillator circuits than in others. The first consideration is the effect that a change in Q, has upon the excitation voltage. Normally, an increase in Qe means an increase in excitation, but this is not true in every case, even in the conventional parallel-resonant circuits. In these oscillators, the feed-back network may con- sist of a crystal unit shunted by one capacitance and in series with another. Referring to figure 1-109, assume at first that the capacitance, C,, shunting the crystal unit in both (A) and (B) is negligible. The generalized circuits are thus equivalent to that of figure 1-108 (B). X, of figure 1-108 (B) is represented by (Xi, -f- Xii>) and by (X,c X,d) in circuits (A) and (B), respectively, of figure 1-109. Referring now to figure 1-109 only, the crystal unit in circuit (A) is connected be- tween the control grid and cathode, so that the principal activity consideration is to obtain the desired excitation voltage across the crystal unit with a minimum power dissipation in the crystal unit. Similarly, in circuit (B) , the higher the crys- tal quality, the less the crystal power that would be required to obtain a desired excitation voltage across Xic. As a first approximation, assume that X, is much greater than R„ so that the voltage across the crystal unit can be assumed to equal IgXe, where I, is the feed-back current and where 1,X„ is 180® out of phase with the voltage across the series reactance, X,., or Xj,, as the case may be. In circuit (A), the excitation voltage is thus equal to I,Xc, and the crystal power dissipation is 1,’Re. If the ratio of the r-f output voltage of the tube to the excitation voltage, Ep/E„ is assumed Section I Eloctricol Paramoterf of Crystal Units to be k, then the ratio of the output voltage to the crystal power. Pc, is : Ep kljXe kQe Circuit (A) pT “ I/R, “ Ig (figure 1-109) The magnitude of the total impedance of the feed- back circuit must be kZ,, where Zg is the grid-to- cathode impedance. As long as it can be assumed that the impedance of the crystal unit is 180° out of phase with the reactance of the series capaci- tance, then the magnitude of the plate-to-grid im- pedance must be approximately (k -f l)Zg. Thus, X in circuit (B), |Xk.| value of Xu. in place of X. in the equation above: Ep kQ, Circuit (B) Pp (k -H 1) Ig (figure 1-109) Figure I- 109. Generalized crystal oscillator circuits, showing two conventional methods for connecting crystal unit (X^, RJ in feed-back circuit WADC TR 56-156 95 Section I Electrical Parameters of Crystal Units The equations for circuits (A) and (B) show that for a given k and a given crystal current, a maxi- mum ratio of E„/Pc is to be obtained when Q* is a maximum. This assumes that the circuit capaci- tance, Cv, directly shunting the crystal unit is neg- ligible. When this assumption cannot be made, the effective Q of the parallel combination must be substituted for Q^. The Q of the combination is approximately equal to Q,, as long as (X, 4- Xe) is numerically large compared with Re, Xy being the negative reactance of Cy. The larger the magnitude of the ratio Xy/X^, the more directly does Qe become the principal activity in- dex. It should be remembered that the direct pro- portionality between Qe and the activity in the example above holds only upon the assumption that Ig is to be held constant, regardless of the value of Qe. Another instance in which the activity of an oscillator is a direct function of Q, would be the unconventional case of an oscillator so de- signed that the crystal unit is operated in series resonance with an external capacitance and with the excitation voltage equal to, or directly propor- tional to, the voltage across either the crystal unit or the series reactance. In such a circuit, use would be made of the resonant rise in voltage that is de- veloped when a component impedance is greater in magnitude than the total impedance. Since the current through the component is the same as that through the total impedance, the step-up voltage ratio is the same as the impedance ratio. At series resonance the total impedance of the crystal cir- cuit would equal Ry (assuming no other resistance in the circuit) , so that if Z, of the crystal unit were approximately equal to X,, the ratio of the voltage across the crystal unit to the feed-back emf would be lyZy/IyRy = Q... The standard crystal units which are intended for use at parallel resonance are tested for operation with definite values of load capacitance, C*. Thus, the recommended oper- ating value of X,. may be assumed to be equal to = |X,I. The maximum value of R* that is per- missible w'ith this value of Xy is also specified. Hence, in the design of an oscillator that must operate satisfactorily with any randomly selected crystal unit of a given type, allowance must be made in the circuit design to ensure that satis- factory activity is obtained for the minimum uCxRy (max))’ MAXIMUM EFFECTIVE Q (Q,J 1-234. Where it is desirable to have an activity WADC TR 56-156 96 parameter that is not a function of the particular external load capacitance, the maximum effective Q (Qra.) offers a convenient index of the maxi- mum potential activity of a crystal unit which is to be operated in a type of circuit whose activity depends primarily upon Qe. The maximum effec- tive Q can be expressed solely in terms of the basic crystal parameters, since the maximum occurs midway between fr and fa, so that af Thus, from equations (1) and (2) of figure 1-98, it can be seen that ^ ^ ^ gxLAfa Re 2R Since Afa Then Qem — [by equation 1 — 208 (1)] 2(_/o 2xfLC yiZ _ 1 Xco 4RCo 4RC„ 4RaiC„ 4R 1—234 (1) Qee, provides a convenient activity factor com- bining all the crystal parameters. It is equal to the maximum step-up voltage ratio that can be ob- tained by operating the crystal in series with a negative reactance. Where another capacitor, C,, is shunted directly across the crystal unit, the maximum effective Q of the combination becomes VLC/4R(C„ -h Cv). FIGURE OF MERIT, M 1-235. In paragraph 1-216, it was shown that the minimum fp obtainable with a crystal unit occurs X when the apparent Q. of the motional arm, is equal to 2 ; that is, unless Rx, the effective resist- ance in the shunt arm, is significant. Within the frequency range at which the crystal unit appears as a positive reactance, the maximum value of Q. occurs at the antiresonant upper limit. This maximum theoretical value of Q, has been selected as a convenient figure of merit to indicate the rela- tive activity quality of a crystal unit, and has been assigned the symbol M. In general, the larger the value of M, the less will be the feed-back energy required to sustain a given activity. If M is less than 2, the crystal unit cannot exhibit a positive reactance, and hence cannot be used in conven- tional oscillator circuits. To sustain oscillations at a desired level, an oscillator will require that the crystal exhibit some minimum value of Q„ equal to 2 or greater, depending upon the oscillator, so that a knowledge of the M of a crystal unit is of Value in determining whether or not the crystal can be used. Formulas for M are : M = X„ 4xLAf. 2xfLC R R RC„ Q ^ y/ns r RC„ 4 Qem 1—236 (1) where Q is the series-arm Q and r = C„/C. Note that M is equal to four times the maximum Q, of the crystal unit, so that the measurement of either will indicate approximately the same performance characteristics. Actually, as M approaches 2 the value of Q,™ as given by equation 1-234 (1) be- comes unreliable, because of the approximations made in its derivation. If M = 2, Q™ is zero, although its approximate formula would indicate a value of 0.5. In practice, however, crystal units with such low values of M are normally far below specified standards, except possibly in the case of v-h-f crystal units operating on harmonics higher than the fifth, so that Q,m, which can be meas- sured directly as the maximum step-up voltage ratio obtainable with the crystal unit in series with a capacitor, provides a reasonably accurate M indication of-^. M was originally chosen as a fig- ure of merit l^cause it can be shown to be a con- stant of proportionality in the equation for Q„ and because it is expressible in terms of the crys- tal parameters alone. As performance parameters of a crystal unit, M and are practically equiv- alent, but M is the parameter more commonly en- countered in treatises discussing crystal activity. PERFORMANCE INDEX 1-236. The fact that the Q, of a crystal unit is the most direct factor influencing the activity first be- came apparent through consideration of the re- quirements necessary for oscillations to build up in the generalized oscillator circuit in figure 1-108 (C). When X, and R, are assumed to be the re- actance and resistance of an actual coil, it can be rigorously shown that oscillations build up as long as (CxpI < Here, Cx and p are both functions of the external circuit, and L, and R, can be as- sumed to be constants of the coil. As the amplitude of oscillations increases, the plate resistance of the tube increases, which in turn causes p to increase. ( Cx may also vary, but usually to a much smaller Section I Electrical Parameters of Crystal Units degree.) Multiplying both sides by <«: |uC,p| or < Q«. From this point of view it would appear that the change in p/Xi which must be undergone before equilibrium is reached, or, equiv- alently, the rise in amplitude necessary to bring the plate resistance to the equilibrium point, will always increase or decrease with Qe, and that with a given R„ the amplitude increases or decreases with Xp. These implications can be misleading. First, with a given Xj, X, is no longer a significant variable if Q,. is equal to 10 or more, but must remain equal in magnitude to X,. In this event the only variable of the activity is Rp. Secondly, the rise in amplitude is more accurately a function of the difference between Qp and the starting value of rather than of the magnitude of Qp alone. Furthermore, Qp is, itself, a function of X,, for as Cx is varied from a relatively large value of capacitance and made to approach zero, Xp must increase with X,. However, Qp does not increase indefinitely with Xp, but reaches its maximum value, Qp,„, when Ci — C„ and then steadily de- creases as Cx becomes smaller. Yet the difference between Qp and the starting value of does continue to increase even though Q„ has passed X its maximum, for the change in is less than the change in the value of |pwC,|. Now, C, and even more so, C, (= Co -f C,), can be considered rea- sonably constant parameters as compared with p during the build-up of oscillations. Referring to figure 1-108 (F), it can be seen that oscillations build up as long as ^ > jp(for the same reasons X that hold in the case of and p/X^). Since Xt Xvp can be assumed to be relatively constant, both sides of the function can be multiplied by Xt, so that X X I it can be said that as long as | ^ = Zp > |/»|, the amplitude of the oscillations continues to in- crease. Thus, for a given starting value of p and with Xt relatively independent of the amplitude, the most direct index of the activity is the equiva- lent parallel-resonance impedance, Z,„ of the gen- eralized oscillator circuit. For this reason, Zp is called the Performance Index and has been given the symbol PI. PI, unlike M and Qp,„, is not a parameter of the crystal unit alone, but of the crystal unit effectively in resonance with' some specified load capacitance, C,. Military S andard crystal units intended to be operated at parallel resonance have a recommended load capacitance WAOC TR 56-156 97 Section I Electrical Parameters of Crystal Units specified. PI meters have been developed for meas- uring the performance index directly, but there are very few such meters available. Where the PI of a crystal unit is desired, it can readily be com- puted from measurements made with standard Cl meters. Various expressions of PI are given below : PI = ^ = 1 ^ LC R a," (Co -f- C.)" R RCt® I Xco ^ I MXco I Note that PI is not a function of the crystal alone, but of C, as well, and care should be taken that the capacitance ratio of C,/C„ is not mistaken for r = Co/C. It can be seen that the maximum PI occurs at antiresonance, where Ci = 0, so that (max) PI = |MXco|. 1-237. The PI of a crystal unit, or more properly, of a crystal circuit is usually found to be an im- portant parameter entering the equations of an oscillator circuit, particularly when the equations express the conditions required for a given out- put. As a simple example, consider again the two generalized oscillator circuits in figure 1-109. In paragraph 1-233, it is shown that the ratio of to crystal power, P^, is equal to kQe/Ig for circuit kQ (A), and equal to ‘ , for circuit (B). As long as Ij is considered predetermined, the volt- age-to-power ratio is primarily a function of k and Q,.. Normally, however, it is not Ig that is to be predetermined, but rather the crystal power that must not exceed the maximum value. When Z, X,., the magnitude of the impedance of the feed-back arm in circuit (A) can be assumed to be equal to kX^, and to kX, k 1 is thus equal to Ep/kXe in (A) and to in (B). On substitution in the Ep/P^ equations: k^QeXe in circuit (B). Ig E„(k + 1) kX. or Ep/Po EpVPc = Ep R» , Ep^ k^PI Similarly, — = = k^PI Circuit (A) Circuit (B) Inasmuch as the power output of the oscillator can be assumed to be directly proportional to Ep*. then, for a given drive level of the crystal unit, the power output will vary directly with the PI. It should be noted that the EpVPc equations above are not affected by the assumption of shunt ca- pacitances, Cy, across the crystal unit. Just as X * X * PI = " 5 ^ — the same value holds even if the XVe shunt capacitance is assumed to be increased by Cy. Frequency Stabilization Quality of Crystal Unit 1-238. The over-all frequency stability of a crystal oscillator is dependent upon the stability. of all the parameters influencing the crystal circuit; these in turn are dependent upon the stability of the power source and the load, as well as the ambient conditions under which the oscillator is required to operate. The over-all stabilizing ability of the crystal is dependent upon both the stability of the crystal parameters when the crystal is exposed to changes in temperature or drive level, and the ability of the crystal to minimize the change in frequency that is necessary when the parameters of the external circuit deviate. It is this latter quality of the crystal that makes the crystal os- cillator superior to oscillators that use only coils and condensers to control the frequency, and is the type of frequency stability that concerns us now. 1-239. The frequency stabilizing property of a crystal is normally expressed as the rate at which its reactance changes with frequency. In figure dX 1-110 are shown the curves of -rj for the series- dt arm parameters L and C. Resonance happens to occur at the frequency at which Xc is changing at the same rate as Xf Since the rate of change of Xi, is a constant, at frequencies i jar resonance it can be said that the total change in reactance with frequency is primarily a function of L, for the absolute rate of this change is the same whether C is large or small. Normally, however, it is not the absolute change in reactance that is important — it is, rather, the change in reactance per percentage change in frequency, or, more usu- ally, the percentage change in reactance per per- centage change in frequency. When the frequency stability is expressed in percentage, it is no longer primarily a function of L, but becomes dependent upon the other crystal parameters as well. Only where the major concern is to produce a definite shift in reactance or frequency for a given change in the external circuit does the major attention center upon the parameter L. Just as the relative activity potential of a crystal depends somewhat upon the type of circuit in which it is used, so also does the frequency stabilizing characteristic of a crystal depend upon the external-circuit de- WADC TR 56-156 98 f, TO U Figure I* 1 10. Rtites of change of reactance of equiv- a/ent series-^rm parameters, I and C, with frequency sign. A relative stability index will be discussed briefly for each of three general types of circuits ; where the crystal is operated at its normal series- resonant frequency, where it is operated in paral- lel with a negative reactance, and where it is operated in series with a negative reactance. FREQUENCY STABILITY AT SERIES RESONANCE 1-240. Since the total series-arm reactance at f, is equal to zero, it is not convenient to express the relative frequency stability in terms of the per- centage rate of change in reactance. Approxi- mately the same considerations apply for the reso- nance frequency, f,. Also, the effective stability in a given circuit may well depend more upon the rate of total impedance change or the rate of phase shift with frequency than upon the actual rate at which the reactance changes. Suppose, for exam- ple, that the feed-back energy must pass through the crystal unit and return to the oscillator input in a certain phase. If, because of a change in the circuit parameters, the feed-back energy is re- turned slightly out of phase, the frequency will have to shift away from the normal resonant point exactly enough for the crystal to correct the change in phase. If the change in phase has orig- inally been caused by a change in the reactance of, say, a capacitor connected directly in series with the crystal, it is only necessary for the frequency to shift the amount necessary for the crystal re- actance to exactly counteract the change in the series reactance. In this case, the resistance of the crystal circuit is not effective in degrading the sta- bility. It is true that the greater the resistance that the crystal faces, the greater must be the fre- Section I Electrical Parameters of Crystal Units quency change to produce a given phase shift. But on the other hand, since the series capacitance faces the same resistance, the initial phase shift due to a change in the capacitance is correspond- ingly reduced. In this case, the frequency stability of one crystal as compared with that of another depends almost entirely upon its relative rate of change of reactance. At series resonance the fre- quency stability factor can be defined as j Y F, = — = 2coL = 2 VlTC 1-240 (1) df where F, is the rate of change of reactance per fractional change in frequency. Thus, in compar- ing one crystal unit with another, the one with the larger L/C ratio can be assumed to provide the greater frequency stability at series resonance. However, if the change in reactance occurs at a point in the circuit only loosely coupled to the crystal, the resistance of the feed-back circuit is relatively ineffective in reducing the phase shift of the feed-back voltage, but instead, tends to in- crease the change in frequency necessary for the crystal to correct the phase. In the case of a feed- back network where the crystal must compensate for a change in phase that is relatively independ- ent of the resistance in the crystal circuit, the fre- quency stability is more directly measured by the rate of change of phase in the crystal circuit as a whole than by the rate of change of reactance alone. 1-241. Figure 1-111 shows that a small phase displacement, A(9, at series resonance is approxi- mately equal to aX,,/R.., where aX, is a small © Zc ' Sc +iO(WHEN ( = fr ) @ Zo = Rc +jAX,(WHEN f = fr +Afl Figure I-III. Phaser diagram, showing change in reactance, AX,, of series-mode crystal required to produce a small change in phase, AS, where the re- sistance of the crystal circuit is equal to R,, WAOC TR 56-156 99 Section I Electrical Parameters of Crystal Units change in the effective reactance of the crystal, and R,. is the total effective resistance in the crystal circuit (equal to R^ of the crystal plus R, of the external circuit). For convenience, the frequency is usually expressed in terms of angular frequency, lu = 27ff radians per second, instead of cycles per second. In equations (1) and (2) of figure 1-98, it can be seen that for small values of Af, the de- nominators of the approximate equations of X, and Re are approximately equal to unity, in which case Xe !=» 2 LAco and R, «=■ R. Thus, ^ „ AXe 2LAca> ^ ~X~ " R + R. The frequency-stability index can be defined to be dw R -j” Rj 1—241 (1) Expressed as the change in phase angle per percentage change in frequency, equation (1) becomes: axid __ 2<»)L 100 dw "" 100 (R + R.) or more simply: wdS du 2coL X 2X, 2 ^ 1—241 (2) The last term on the right shows that where the fractional rate of change is concerned, the fre- quency stability is directly proportional, not sim- ply to L, but to the square root of the L/C ratio. Equation (2) also shows that the frequency stability is inversely proportional to the crystal circuit resistance. But it must be remembered that this is true only to the extent that the original phase shift of the input to the crystal circuit can be considered independent of Re- 1-242. As an exaggerated example, we can see that minimum stability is to be expected if the input to a high-resistance, series-resonant, feed- back circuit is supplied through a weak coupling from a plate tank circuit sharply tuned to the reso- nant frequency but having an impedance small compared with the Rp of the tube. Since a slight change in the parameters of the tank circuit could shift the phase of the feed-back input almost 90 degrees, such an oscillator would obviously be completely unstable, even if it were assumed that oscillations could be maintained. FREQUENCY STABILITY AT PARALLEL RESONANCE 1-243. When it can be assumed that a crystal unit WADC TR 56-156 100 is effectively operating in parallel with an external capacitance, C„ the value of which is relatively independent of small changes in the frequency, the frequency stability can be assumed to be di- rectly proportional to the rate of change in the reactance of the motional arm of the crystal. In this case, it would seem that a frequency-stability factor for parallel resonance Fp = ^ ■ 0, = 2 ViX = F, do) would be appropriate — ^just as in the case of series resonance — and Fp would be identical with F,. However, it will be recalled (see figure 1-103) that the higher the reactance, i.e., the smaller the value of Ct, the less stable will the oscillator become. Taking this into account, a more accurate indi- cation of the stabilizing quality of a parallel-reso- nant crystal is given by what is called the fre- quency-stability coefficient, which is the percent- age rate of change in reactance for a percentage change in frequency. Thus, dX. 100 _ 2a)L f ^ " d« ■ X. ■ 100 “ X. Af This result is quite interesting, for it indicates that for crystal units of the same frequency equal sta- bilities can be achieved simply by operating the crystal units at the same value of Af above series resonance. Since the same equation holds for any value of L and C, it is not immediately apparent as to why a crystal is so much more stable than a conventional inductor and capacitor. In para- graph 1-208 it was found that the ratio is Q equal to If Afp is substituted for Af„ and Ct is substituted for C,, then Fx. = f 2 Ct Afp " C It can be seen that in order for a conventional series-parallel inductor-capacitor network to have the same theoretical stability as a crystal, the shunt capacitance must be thousands of times greater than the series-arm capacitance. The series arm of such a network would require an extremely small L/C ratio. The parallel impedance would be small, and the net series-arm reactance much smaller still. The crystal, on the other hand, has such a small value of C that the reactances are reasonably large even for small values of Af. Al- though the equation for the frequency-stability coefficient indicates an unlimited stability if Afp is simply made small enough, this would be theo- retically true only for a circuit resistance equal to zero. In pratice, as X, approaches the motional- arm R in magnitude, a given change in Xt will cause a greater change in the phase of the over-all circuit impedance than will the same change in X.. Of significance is the fact that the frequency- stability coefficient, Fx., represents the stabilizing effect of a crystal for the percentage change in the reactance of the total effective shunt capacitance, Ct (= Co -f- Cx). With C„ considered constant, a given percentage change in the total capacitance becomes less than the actual percentage change in the variable component, which we can assume to be the equivalent external capacitance, C,. For a given Ct, the larger the ratio C„/Ci the smaller will be the percentage change in Ct for a given percentage change in Cx, and the greater will be the oscillator stability in the face of changes in the external circuit. In other words, the effective frequency-stability coefficient of the crystal unit as a whole is greater than that of the motional- arm alone if it can be assumed that the percentage changes in C„ will be negligible compared to those in C,. When R is small compared with X„ X^ «== XcoX,/(Xro + X,) and the effective frequency- stability coefficient becomes: ^ ^ 2LXco^ a»(Xco -f X.) do, ■ X. (Xco + ■ Xco X. 2ci>Ij Xco fCx ^ X ■ Xc„ -t- X. ^ AfX Since then Ct 2Ct* Fx. = Fx. • ^ 1-243 (1) It should be remembered that equation (1) is based upon the assumption that C, is constant and that any change in X, will be due to a change in C,. If Co is effectively increased by a fixed ca- pacitance, Cv, directly shunting the crystal unit, the effective variable C, becomes smaller. The effective C„ insofar as the frequency stability is concerned, will equal Ct — (C„ -f- C, ) . Substituting this effective value of C. in equation (1) will pro- vide a more accurate frequency-stability coefficient for a crystal unit directly shunted by a fixed C,. Section I Electrical Parameters of Crystal Units FREQUENCY STABILITY AT SERIES RESO- NANCE WITH EXTERNAL CAPACITANCE 1-244. The effective frequency-stability coefficient, 20r* Fx,. = provides an appropriate index of the frequency-stability quality of a crystal unit oper- ated in series resonance with an external capaci- tance Cx, for the same reasons that make the co- efficient applicable in the case of parallel-resonant circuits. Ct, here, represents the sum of two actual capacitances, Co -f- C„ and has a more concrete meaning than simply a generalized parameter. Fx. gives the percentage change in X,. per percentage change in frequency. The reciprocal, 1/Fx„, can be interpreted as equaling the percentage change in frequency that will occur per percentage change in the negative reactance Xj. In unconventional circuit designs, where a significant phase shift can occur as a result of changes in the impedances in the oscillator output circuit {which is only weakly coupled to the feed-back input), the resist- ance of the feed-back circuit may need to be taken into account in a manner similar to that discussed in the case of crystal units operating in series resonance. FREQUENCY STABILITY OF OVER-ALL CIRCUIT. 1-245. Although the absolute values of the fre- quency-stability indices discussed in the foregoing paragraphs depend upon generalized parameters of the external circuit, the values are primarily useful for indicating the relative stabilization qual- ity of one crystal unit as compared with another when operated under similar circuit conditions. The actual frequency drift due to changes in the supply voltages, tube characteristics, circuit im- pedances, etc., depends upon the particular oscil- lator design as well as upon the performance characteristics of the circuit components. The percentage variation that can be expected in the generalized parameter, Cx, is of equal importance in gauging the frequency stability of the oscillator as a whole. The crystal-unit frequency-stability coefficients appear as single parameters among others in the frequency-stability equations for each particular type of oscillator circuit. In general, the series-resonant type of oscillator has the greater frequency stability, permitting tolerances from four to twenty times as narrow as those normal for parallel-resonant oscillators. Indeed, in a well- designed series-resonant oscillator where the re- active components are negligible in their effect on the phase of the feed-back voltage, the frequency stabilization of the crystal unit can be very nearly WADC TR 56-156 101 S«ction I Electrical Parameters of Crystal Units perfect, so that the most significant factor to con- sider is the stability of the equivalent-circuit parameters of the crystal unit, itself. One source of frequency instability common to both series- resonant and parallel-resonant oscillator circuits is the presence of harmonics in the output. For cer- tain applications, such as in crystal calibrators, these harmonics are desirable, but in most cases it is preferable that they be kept to a minimum. Harmonics are unwanted not only for the sake of a sine-wave output as such, but also because they introduce reactive components in the crystal cir- cuit, thereby increasing the chances of frequency instability. The harmonics can be reduced by de- signing the oscillator plate circuit to provide a low-impedance bypass path for them, and by using low plate and grid voltages. Unwanted reactive effects in the oscillator circuit also occur as a result of feedback from the amplifier stages following the oscillator. These can be minimized by the use of proper shielding, buffer amplifiers, neutralizing circuits, and by careful attention to the physical layout in designing the equipment, to ensure that all leads are as short as practicable and that the oscillator is electrically isolated from circuits carrying high amplitudes of r-f voltage or current. The effective load capacitance, Cj, with which the crystal unit resonates is usually a function of the vacuum-tube parameters, the load resistance, the effective grid resistance, as well as the reactive impedances in the feed-back and output circuits. All these variables are, in turn, functions of the oscillator output and the grid and plate d-c volt- ages. Thus, the frequency stability is dependent upon the degree of voltage regulation, the con- stancy with which the oscillator load is main- tained, and in the care taken in the original design to ensure that the circuit components are so pro- portioned that the effects of variations in the tube parameters are minimized. Silvered mica capaci- tors normally are to be preferred for fixed capaci- tances in the tuned circuits. Those capacitors hav- ing dielectrics composed of titanium compounds can be used for r-f bypass purposes, but are too variable under changes in temperature and voltage for use as tuning components. Air-dielectric ca- pacitors are almost always adjustable. With the exception of the vacuum-dielectric capacitor, the air-dielectric type is the most stable and is to be preferred for small capacitances and variable tun- ing elements. As a rule, the improvements in circuit design that permit of greater frequency stability necessitate additional circuit components, additional tuning adjustments, narrower operating frequency ranges, smaller voltage or power out- puts, or some combination of the above. Bandwidth and Selectivity Parameters of Crystal Unit TOE CAPACITANCE RATIO, r 1-246. The bandwidth of a crystal unit refers to the particular frequency range over which the crystal unit can be operated in a given oscillator, filter, or transducer circuit. In the case of a con- ventional oscillator circuit, the applicable fre- quency range is that in which the crystal can appear as an inductive impedance. In cycles-per- second, this range is Af, = f. — f,. Percentage- wise, the bandwidth is , which, as shown Ip in paragraph 1-208, is approximately equal to ^ Although the practicable operating range does not extend over the entire band, it can be seen that the relative merit of a crystal unit insofar as its range of frequency adjustment is concerned can be indicated inversely by the param- eter r, whereas the relative selectivity is indicated directly by r. In figure 1-95, it can be seen that the smallest theoretical values of r (when the dis- tributed capacitance is negligible, so that r = r,) are obtained with the low-frequency, length- extensional-mode elements of the X group. The smallest capacitance ratios are provided by ele- ment E, which has values of r as low as 120 to 125. These are equivalent to a resonance-to-anti- resonance bandwidth on the order of 0.4 per cent of the nominal frequency. For the high-frequency A and B elements, the bandwidths are approxi- mately 0.2 and 0.083 per cent, respectively. 1-247. Insofar as frequency control is concerned, the resonance-to-antiresonance bandwidth is impor- tant primarily as a relative index of the frequency range through which a parallel-resonant oscillator can be made to operate by varying the load capaci- tance, Cx. For example, the tuning adjustments of an oscillator employing an A element can vary the frequency approximately two-and-a-half times as much as can the same adjustments if the oscillator employs a B element. Although small frequency adjustments are possible, the high selectivity of quartz crystals precludes their use in frequency- modulated oscillators. Eventually, it may be that crystal units mounting high-frequency EDT plates, which have capacitance ratios as low as 20, will find an application in this field, but at the present time EDT crystals are used almost exclusively in filter networks. As a filter element, the capacitance ratio of a crystal is of greater importance than in frequency-control circuits. Filter networks, com- posed of crystal units alone, can be designed for a WADC TR 56-156 102 Section I Electrical Parameters of Crystal Units maximum pass band of - ■ , which in the case Ip of quartz means a maximum pass band of 0.8 per cent. For the low-frequency networks, such as are normal to telephone carrier systems, this is much too selective for passing voice channels. For this Peason, quartz crystals employed in 1-f telephone carrier filters must be used in conjunction with inductors and capacitors. The narrow bandwidths of quartz elements used alone are primarily appli- cable in filters when it is desired to pass a single frequency, such as the pilot signal of a carrier system. ELECTROMECHANICAL COUPLING FACTOR, k 1-248. To the extent that the equivalent circuit of figure 1-91 is applicable it can be assumed that when a crystal unit is connected across the ter- minals of a battery the ratio of the energy stored in electrical form to the energy stored in mechan- ical form is equal to the capacitance ratio r = Co/C. The electrical energy is that stored in the static capacitance, Co, and is equal to V’Co, where V is the applied d-c voltage. The mechanical energy is the energy that is stored because of the piezoelectric strain in the crystal, and is equal to 1/2 V*C. In transducer applications, it is useful to rate a crystal according to the ratio of stored mechanical energy to total applied electrical energy under the conditions of d-c or very-low-frequency applied voltages. The parameter for this purpose is the electromechanical coupling factor, k, equal to the square root of the ratio of the stored mechanical to the total input energy. As such, k is an index of the crystal efficiency as a transducer. This factor is given by the formula k = 1—248 (1) where c is the dielectric constant, s is the elastic compliance, and d is the piezoelectric constant giv- ing the ratio of strain to field. According to the energy ratio, k* should equal 7 ; — or approxi- O© “P V 1 c mately -. However, the capacitance ratio, at resonance can be shown to be ^ times as large as the theoretical ratio at zero frequency. Actually, then the ratio is stored mechanical energy total stored electrical energy !L“ 9 . 8C. 1—248 (2) where C and Co are the equivalent capacitances at resonance. Since the bandwidth is proportional to C/C„, so also is it proportional to k*. In transducer applications, when an inductor is shunted across the crystal to tune out the electrical capacitance, and the crystal is operated near resonance, up to 90 per cent efficiency is possible in the conversion of electrical to mechanical energy. Under these conditions, k is not a direct index of the transducer efficiency, but it does serve as a parameter for estimating the frequency range over which the efficiency is 50 per cent or greater. The ratio of the highest to the lowest frequency for greater than 50 per cent conversion is: Crystal Quality Factor, Q 1-249. The quality factor of a crystal unit is the Q of the motional arm at resonance. Thus, Quartz crystal units are obtained with Q’s rang- ing in value from 10,000 to more than 1,000,000. The Q is a performance parameter that provides an indication of the ratio of the stored mechanical energy of vibration to the energy dissipated in the crystal unit per cycle at resonance. If I, is the r-m-s current through the series arm at resonance, then, at the instant the current is a maximum, the equivalent capacitance C can be assumed to be completely discharged and all the vibrational en- ergy, Ev, is at that instant in kinetic form. This energy is equivalent to that stored in motional-arm • inductance, L. Therefore, E. = = CL 2 1—249 (1) The energy dissipated per second, P^., is I.’R. Thus, the ratio of the stored mechanical energy to the energy dissipated per second is E, _ I.*L _ L Pe ~ I."R “ R 1—249 (2) It can be seen that for a given wattage, the greater the L/R ratio the greater will be the amplitude of vibration. Regardless of the wattage, for a given L the amplitude of vibration will vary approxi- mately directly with the current. Theoretically, WADC TR 56-156 103 Section I Electrical Parameters of Crystal Units since there are no Military Standards setting a minimum limit for the series-resonance resistance, a crystal unit can be so excellently mounted that it would be vibi-ated near its elastic limit if atten- tion were given only to the power dissipation rather than to the current. Such a situation is not likely to arise except possibly in the case of a crystal-controlled power oscillator, where space and cost limitations require a crystal drive level far in excess of the rated level. More important from the point of view of maintaining a sinusoidal wave shape of the excitation voltage and of im- proving the stability of the oscillator is the ratio of the stored energy to the energy dissipated per cycle, rather than per second. In terms of angular frequency, the dissipation per radian is I.’R/w, so that Ey _ P./u) " I/R 1—249 (3) In an actual series-resonant circuit, it is the Q of the entire circuit rather than of the crystal unit alone that must be considered, so that R should be replaced by the total circuit resistance. If a tuned, class-C-operated circuit is to be effective in maintaining a sinusoidal wave shape and in re- ducing harmonics, the energy stored in the circuit should be at least twice the amount that is dissi- pated over the entire cycle. That is. (min) Ey Pc/f 2r 2 This requirement is met easily in quartz-crystal circuits, but it is an important consideration in the design of plate tank circuits that are to be fed in pulses not smoothed by the action of the crystal. The crystal Q is also an important parameter in crystal filters. In general, the higher the Q the sharper the pass band. 1-250. Since the Q of a crystal is equal to and since the ^ ratio for a given frequency can be increased to almost any value desired by de- creasing the electrode area and by orienting the crystal in a direction of weak piezoelectric effect, or by using twinned crystal blanks, it might be wondered why much larger values of Q are not in use. The reason is that the L/C ratio and the equivalent series-arm resistance of the crystal are not independent of each other. As \/L/C increases, so also does R. This can be intuitively seen if it is kept in mind that fundamentally the Q is the ratio of the energy stored to the energy dissipated per angular cycle. Suppose that we have two crystal plates, A and B, both of approximately the same size and normal frequency, and both mounted exactly alike in that the frictional losses of one are the same as those of the other for the same energy of vibration. We shall also assume that these mechanical losses account for most of the crystal driving power. In other words, we are assuming that the two crystals have approxi- mately the same quality factor, Q. Now, suppose that crystal A has a much smaller electrode area than does crystal B, or that for some other reason the piezoelectric effect of A is very weak com- pared with that of B. Under these conditions, crystal A will have a much larger equivalent L/C ratio than does crystal B. But since the Q of A equals the Q of B, it can be seen that the series- arm R of A must be greater than the series-arm R of B in the same proportion as the square roots of the respective L/C ratios. It should be under- stood that the Q and the L/C ratio are compara- tively independent variables as far as R is con- cerned. Where R could not be estimated without a knowledge of Q and L/C, the latter theoretically could be approximated separately and independ- ently by an examination of the fabrication of a crystal unit. L and C, for example, are approxi- mately predetermined by the electrode area and the type and size of the crystal element. The Q is also to a certain degree a function of the same variables, but for given internal frictional prop- erties, is primarily determined by the quality of the crystal finishing and mounting. Thus it is that the Q is largely determined by the frictional losses and is not subject to control by varying the L/C ratio. Indeed, as the L/C ratio increases, the piezo- electric effect can become so weak and the resist- ance so high that the crystal cannot be shocked into oscillation unless very high voltages are em- ployed. Once in oscillation, a high L/C crystal unit could presumably operate satisfactorily, except that only very small currents could be withstood without the cr.ystal shattering or arcing. There is a hypothetical case where an exceptionally large L/C ratio could be practical. Such a situation would arise if for any reason the external circuit resistance faced by a crystal could not readily be reduced below some large minimum value. In this event, the use of a crystal unit of normal Q but large L/C ratio would prevent the over-all circuit Q from being excessively degraded by the external resistance. Ordinarily the selection of a crystal unit will be made on the basis of considerations other than the L/C ratio, but where all else is equal, including the average values of Q, it might be assumed that the crystal units having the some- WADC TR 56-156 104 what higher values of L/C are generally more suitable for those oscillators which do not require a crystal to sustain oscillations, but only to stabi- lize them. Such a circuit oscillating at or near the crystal frequency can build up the crystal vibrations over a large number of cycles of small amplitude, thereby obviating the need of large voltage surges or abnormally high vacuum-tube amplification. Stability of Crystal i*arameters 1-251. Regardless of how well designed a crystal oscillator may be, or how high the degree to which the crystal stabilizes fluctuations in the external circuit, the over-all performance will depend upon the stability of the crystal parameters, themselves. Changes in the crystal parameters are primarily due to aging, changes in the ambient temperature, spurious modes, and to changes in the drive level. Aging, here, is used in its broadest sense to include practically any nonreversible change in the crystal characteristic from whatever cause. The principal causes and effects of aging are discussed in para- graphs 1-172 through 1-181. EFFECT OF TEMPERATURE UPON CRYSTAL PARAMETERS 1-252. The temperature-frequency characteristics of quartz plates are covered in the description of the various elements, and- will not be repeated here except to note that a change in the frequency means a change in the LC product of the motional arm. To what extent the frequency drift may be due to a change in L and to what extent to a change in C would require very precise measurements of f, and fp, and. the approximate formula in para- graph 1-225 for computing C from the measured parameters would need to be replaced by a more rigorous equation. Although of theoretical value, such small changes in L or C are not, in them- selves, of practical importance in circuit design — rather it is the change in the LC product (i.e., in the frequency) which is important, and which must be kept to a minimum. Low-temperature- coefiicient crystals have been developed for this purpose, but only the GT, at low frequencies, and the AT, to a lesser extent, at high frequencies provide a near-zero coefficient over a wide tem- perature range. The more exacting the require- ments, the more expensive the crystal unit will be. Fortunately, zero coefficients can be obtained at different temperatures by slight variations in the orientation angle of the cut. By mounting the crys- tal in an oven thermostatically controlled near the zero-coefficient point of the crystal, the tempera- Section I Electrical Parameters of Crystal Units ture effected frequency deviation can be kept very small. Indeed, an ideal oven having a zero tem- perature fluctuation would permit any type of quartz cut to be stable provided the drive level remained constant. Nevertheless, the use of an oven is to be avoided where possible, because of the additional cost, space, weight, and power re- quirements, and also because the crystal pins of the oven increase the shunt capacitance across the crystal. The additional shunt capacitance proves increasingly objectionable at the higher frequen- cies, and makes it necessary that either the oven dimensions be as small as possible or that the entire oscillator be mounted within the oven. Either requirement serves to reduce the stability of the oven temperature, particularly if the am- bient temperature varies between wide extremes. For ovens of practical size and construction, some frequency deviation is to be expected as a result of temperature changes. If this deviation is to be kept to an absolute minimum, precise tempera- ture-coefficient characteristics must be specified in selecting a crystal unit, or a greater precision in temperature control than is now attainable in the average crystal oven must be sought. An in- genious method of obtaining practically a zero temperature coefficient for A elements over a span of 20°C and more is being developed by the Hunt Corporation. During a luncheon conversation several years ago between E. K. Morse, S. Ryesky, and D. Neidig (the former, a Government repre- sentative, the latter two of Hunt) concerning the possibilities of improving the frequency stability of Radio Set AN/ARC-1 in its first modification, the idea originated of operating two temperature- compensating equal-frequency A elements in series. The angles of cutting could be so selected as to provide equal temperature coefficients of op- posing polarities which would cancel when both crystals were operating at the same temperature. However, little was attempted in this field until recently. Experimental models show^ that over room-temperature ranges the frequency deviation can be quite negligible. Aging data* is still insuffi- cient, but over a period of six months a stability of about one-half part per million has been achieved, with three-fourths of the drift occurring in the first three months. Probably the most sig- nificant recent activity in the development of fabrication processes designed to stabilize the crystal parameters against changes in tempera- ture centers around the current investigations under the direction of Dr. E. A. Gerber of the Signal Corps Engineering Laboratories. As re- ported by Mr. D. L. Hammond in a modest paper. Effects of Impurities on the Resonator and Lat- WAOC TR 56-156 105 Section I Electrical Parameters of Crystal Units tice Properties of Qvurtz, presented at the 1955 Signal Corps Frequency Control Symposium, a systematic exploration is under way to discover and catalog the effects on the parameters of quartz crystals which have been synthetically grown to include controlled percentages of impurities. Im- purity elements being experimented with include aluminum, boron, calcium, germanium, lead, se- lenium, tin, titanium, and zirconium. This work undoubtedly has revolutionary possibilities. The discoveries already made presage the probability that temperature effects, which are now so im- portant a problem, can in the future be largely eliminated by growing crystals, for particular cuts, with controlled impurities of proper quanti- ties and proportions. 1-253. A crystal operated at an overtone mode will have temperature-frequency characteristics different from those exhibited by the same crystal at its fundamental vibration. For the control of very high frequencies the A element is normally preferred to the B element, because of its stronger piezoelectric effect and because of the smaller fre- quency deviation possible for large variations in temperature. However, an AT cut ideally oriented for operation at the fundamental mode is not usually ideally oriented for the higher modes. A research team at Philco Corporation investigat- ing the characteristics of harmonic-mode crystals found that by far the greatest change in the tem- perature-frequency characteristics of A elements occurs at the first operable harmonic jump, i.e., between the fundamental and the third harmonic. (See figure 1-112.) Since the subsequent changes at the higher harmonics are relatively small, a crystal suitably oriented for the fifth harmonic will usually be suitable for operation under the same temperature conditions at all other over- tones. The sensitivity of the crystal to slight changes in the orientation angle is acute. Figure 1-113, for example, shows the degree by which the characteristic curve of an llth-harmonic A element is rotated by successive changes in the orientation angle of only 3 minutes each. If this crystal were to be operated at room temperature, an orientation of approximately 35°27' would appear to be preferred. For operation under temperature variations of — 55° to +70°C, an orientation of 35°30' permits the minimum total frequency deviation from a room-temperature mean. Finally, if the crystal is to be mounted in an 85° crystal oven, an orientation of 35°33' would be optimum. EFFECT OF SPURIOUS MODES UPON CRYSTAL PARAMETERS 1-254. Closely allied with the problem of tempera- ture control is the problem of avoiding spurious modes. Spurious modes are most apt to occur in the case of thickness-shear crystals. Among these elements, the AC and BC cuts provide the purest frequency spectrum, but unless crystals of these types are provided with precise temperature con- trol their larger temperature coefficients prevent their being preferred over A and B elements. Cut- ting the crystal blank to the proper face dimen- sions is the most important factor in avoiding unwanted modes, but even when due precautions are taken, sudden apparent variations in the mo- tional-arm parameters of individual crystal units -45 -25 -5 +5 +25 +45 +65 +85 CRYSTAL TEMPERATURE .CENTIGRADE DEGREES Figure I-II2. Typical variations in frequency-temperature characteristics of A element when operated at different harmonics \ WADC TR 56-156 106 are hot uncommon. These effects occur most often during variations in temperature, and are due to the fact that the temperature coefficients of nearby modes are quite high. The activity and frequency curves Versus temperature of an erratic A element at series resonance in a tuned bridge circuit are shown in figure 1-114. The activity was measured by the grid current. No tuning adjustments were made during the temperature run. Note that the sudden jumps occur at some of the same frequen- cies, which, at the high-temperature portion of the curve, are apparently of a reasonably pure mode, indicating that the temperature coefficients of the desired and the unwanted modes are differ- ent. It can also be seen that the sudden dips in frequency are accompanied by abrupt changes in f Section I Electrical Parameters of Crystal Units activity, the latter probably being due to higher motional resistances for the unwanted modes. Un- wanted modes are not always accompanied by changes in the resistance. For example, a sudden jump from one frequency to another, but without the dipping effects shown in figure 1-114, where the temperature-frequency curve is effectively broken into two smooth curves, may have very little effect on the activity. This type of frequency jump, which was quite common in the old Y-cut crystals, seems to be due primarily to small de- fects in the finishing of the crystal blank. Where only one such jump occurs during the temperature cycle, it can usually be eliminated by a slight re- tuning of the oscillator circuit. However, retuning the oscillator circuit, particularly if the crystal is CSVSTAI. TEMPERATURE , CENTIGRADE DEGREES Figure 1-113. large clockwise angle of rotation of frequency-temperature curve of harmonic-mode A element caused by small increments (3 minutes of arc) in the cutting orientation angle about the X axis WAOC TR 56-156 107 Section I Electrical Parameters of Crystal Units being operated near series resonance, will have little effect upon those temperature-frequency characteristics due to unwanted modes that are in- herent functions of the major dimensions of the crystal blank. A crystal unit having characteristics similar to those shown in figure 1-114 should not be used where the operating temperature is ex- pected to extend into the erratic region. Unfor- tunately, the specifications for most of the crystal units listed in Section II of this handbook are not rigorous enough to provide a guarantee against unwanted modes for every type of unit, if the effects upon the frequency and the effective resist- ance do not cause over-all deviations beyond the maximum allowed. On the other hand, “jumpy” crystals are the exception rather than the rule, but if particular precautions are necessary where wide temperature variations are to be encountered. . VCTIVITY-l EMPERATl JRE CHAR/ ^CTERISTK "Ny — ' r Y L t CRYSTAL TEMPERATURE, CENTIGRADE DEGREES Figure 1-114. Activity-temperature and frequency-temperature characteristics of harmonic-mode A element, showing effects of unwanted modes WADC TR 56-156 108 i>^liilfi:«liose crystal units should be used which are specified by Military Standards to be free of un- '^an^d modes over the desired temperature range. 1-255. The overtone modes of the thickness-shear eleihei)^ are more likely to be troubled with spuri- ous frequency dips of the type shown in figure 1-114 th^ are the fundamental modes, but a crys- tal tiiat is erratic at its fundamental vibration usually exhibits a pure frequency spectrum at a high harmonic. Indeed, because the frequencies of the unwanted and the desired harmonics do not increase in the same proportion, one method of lessening the probability of interfering modes at the higher harmonics is to deliberately cut the crystal with edge dimensions which favor spurious responses at the fundamental frequency. Never- theless, the overtone crystals have a tendency to oscillate at two or more thickness-shear frequen- cies. Usually, this seems to be due to slight differ- ences in the thickness of the crystal from one point to another. For each order of the harmonic, n, the crystal can be imagined to be divided into n layers perpendicular to the thickness, with each layer be- ing a separate crystal vibrating 180° out of phase with the neighboring layers on each side. If n is an even number, the separate sections tend to can- cel each other’s electrical effects at resonance. For this reason the even harmonics cannot easily be electrically excited. In the case of the odd har- monics, there is always effectively one vibrating layer whose alternating polarity is not neutralized. Most pf the activity is more or less centered in one particular region of the crystal plate. If the thickness at an active point differs slightly from the thickness at a neighboring point, there will be a tendency to jump from one activity center to another, and small jumps in the frequency can result. In the case of crystal plates vibrating at high harmonic modes, a small variation in the thickness dimension is generally more likely to produce a sudden frequency jump than if the same crystal were vibrating at its fundamental mode. If there is little difference between the equilibrium conditions of two vibrating stages, the frequency may shift back and forth at an audio rate, thereby effectively modulating the oscillator output with an audio frequency. Such frequency jumps are best avoided by the use of ceramic-button holders, the design of which concentrates the excitation in a small area at the center of the crystal where the most uniform thickness is attainable. Occasion- ally, it is found that the small frequency jump occurs only at a particular adjustment of the oscil- lator, and therefore it can be avoided by slight changes in the oscillator tuning. Even so, unless Section I Electrical Paramotere of Cryttal Unit* the temperature is to remain reasonably constant, a crystal unit exhibiting any tuning jump at all should not be used. For although an unwanted mode that occurs during a temperature cycle may never appear during a tuning adjustment, the re- verse situation is rarely found — a frequency jump that can be caused by a tuning adjustment is al- most certain to appear during a temperature cycle. EFFECT OF DRIVE LEVEL UPON CRYSTAL PARAMETERS 1-256. There is insufficient data and standardiza- tion at the present time to analyze or to predict exactly the effect a change in the drive level will have on a crystal unit of a given type. Not only do crystal units of the same type exhibit various reactions, depending on the nominal frequency, the method of fabrication, and the manufacturer’s specifications, but even when all these factors are the same for a sample of crystal units, the indi- vidual reactions to changes in drive level are un- predictable. The frequency and series-arm resist- ance curves versus drive level in figures 1-115 and 1-116 are shown as examples. These curves were prepared from data obtained during a Signal Corps research project at New York University by a research team consisting of Messrs. Don J. R. Stock (Director), L. Silver, E. Strongin, A Yev- love, and A. Abajian. The curves in both figures were made from the same set of 9-mc crystal units — AT-cut, electrode-plated, wire-mounted types CR-18/U and CR-19/U, all made by the same manufacturer. FREQUENCY VERSUS DRIVE 1-257. In figure 1-115, note that although the fre- quency of the average crystal unit tends to in- crease with drive level, this effect is by no means to be found at all drive levels for all crystal units. Unfortunately, the temperature-frequency curves for these same crystals are not available, so it is not possible to judge how much of the frequency deviation is due simply to the rise in temperature with drive level. However, the frequencies of other A elements have been tested for frequency devia- tion versus power, and even though the increases of temperature due to drive occur at points of negative slope on the frequency-temperature curve, the actual frequency-drive level curve gen- erally reveals a positive slope. This increase in frequency with drive is apparently due to a rela- tively large temperature-gradient coefficient. The net effect on the frequency is due to the combined influences of the changes in both the temperature and the temperature gradient, which influences WADC TR 56-156 109 Section I Electrical Parameters of Crystal Units Figure 1-115. Frequency deviation versus drive for a random sample of 9-me A elements. All crystal units are the products of the same manufacturer and are similarly fabricated and mounted In HC-d/U holders POWER IN MILLIWATTS Figure 1-116. Resistance deviation versus drive for a random sample of 9-mc A elements. Curves are for the same sample of crystal units whose frequency-drive characteristics are shown in figure 1-115. Correspondingly numbered curves are those of the same crystal unit WADC TR 56-156 110 Section I Electrical Parameters of Crystal Units may or may not be in opposition. From the appear- ance of the curves in figure 1-115, it is possible that those curves starting with a negative slope may be primarily responding according to the nor- mal temperature coefficient. No data is available concerning the degree by which the orientation of the crystal plate relative to the mounting wires might influence the thermal-gradient effect. 1-258. The rise in temperature per milliwatt of drive varies widely with the types of mounting used and the sizes of the crystal plates. For wire- mounted units, most of the heat generated is due to friction at the points where the crystal is sup- ported. With the heat source thus concentrated in a small region of the crystal surface, steep thermal gradients can be expected. The over-all rise in temperature is also greater in the case of wire- mounted units, since most of the thermal-leakage must be through the air, which, like all gasses, acts as a thermal insulator. If the crystal unit is vacuum-sealed, the temperature change per milli- watt may increase by a factor of from two to ten, depending upon the size of the supporting wires and how much of the crystal surface is metal- plated. With the air evacuated, the heat leakage is primarily through the supports and by radiation. The amount lost by radiation depends largely upon the emissivity of the crystal surface, which is approximately 40 times as great for unplated as for plated areas. If it can be assumed that the heat is evenly distributed over the volume of the crystal, the temperature rise of a one-centimeter-square crystal wire-mounted in an HC-6/U holder (not evacuated) can be expected to be approximately 0.3 to 0.4 centigrade degrees per milliwatt of drive. In practice, however, the temperature of the parts of the crystal where most of the heat is gen- erated may increase as much as 10 times this amount. If high or variable drive levels are to be used, pressure-mounted crystal units should be employed. The relatively large contact area be- tween the crystal and the supporting electrodes permits a more uniform distribution of the heat, thereby reducing the magnitudes of the thermal gradients. The pressure mounts also provide a much higher thermal conductivity away from the crystal, thus enabling a much smaller temperature rise per milliwatt of drive. Finally, the pressure mount provides better mechanical and aging pro- tection for the crystal when operated at high- amplitude vibrations. Regardless of the type of mounting, it is never desirable from the point of view of stability or of long crystal life to use a higher crystal drive than absolutely necessary. 1-259. In tests made with GT-cut crystals, where MICROAMPERES PER MM OF WIDTH Figure 1-117. Frequency deviation versus crystal current density for two GT-cut crystals which were sub/eeted to different periods of etching* the frequency deviation with temperature is prac- tically zero over a 100-degree centigrade range, the opportunity has been afforded to study the frequency deviation due to drive alone without the complication of temperature-coefficient effects. Ex- periments with G elements, as reported by R. D’Heedene, reveal a negative frequency deviation with drive, as shown in figure 1-117. Note that the GT plate given a deep etch maintained its stability during much higher drive levels than did the plate etched only 20 minutes. Since the effective resist- ance of the better finished crystal can be expected to be less than, and to be more stable with increas- ing drive than, the resistance of crystal A, the changes in frequency with changes in crystal power may have been much closer than the curves in figure 1-117 indicate if it was assumed that the resistances of the two crystals were equal. If the change in frequency is due primarily to changes in the thermal gradients, it is more directly a function of the crystal power. On the other hand, if the fre- quency deviation is due primarily to mechanical strains resulting from high amplitudes of vibra- tions, it is more directly a function of the crystal current. Although the evidence now suggests that it is the thermal gradients that are the primary factors, certainly a lowering of the frequency can be expected for any mode if the elastic limit is approached too closely. After crystal units are subjected to high amplitudes of vibration, they do not return immediately to their original frequen- cies when the drive is reduced to a low level. A period of days or weeks may ensue before the crystal unit regains its former characteristics, during which time the performance resembles that of a crystal rapidly aging. 1-260. Although the frequency-versus-drive char- acteristics of individual crystal units deviate con- siderably from the norm, the characteristics are generally similar enough to plot reasonably de- WADC TR 56-156 111 Section I Electrical Parameters of Crystal Units pendable average curves when the fabrication processes and the frequencies are the same. Such curves showing average frequency deviation versus power are illustrated in figure 1-118. Each curve represents the average of several samples from a representative manufacturer for a given fre- quency. The curves with the same letter corre- spond to crystal units of the same manufacturer. All the crystals are A elements, metal-plated and wire-mounted in HC-6/U holders. In every case, it can be seen that the average tendency is for the frequency to increase with power. RESISTANCE VERSUS DRIVE : -261. The resistance curves shown in figure 1-116 are more or less typical of the wide variations that must be considered in the design of an oscillator. A minimum performance level must be maintained regardless of the resistance of the crystal unit, as long as the resistance complies with the military specifications. Actually, the average series-arm resistance of the crystal units shown is quite low for 9-mc crystal units. As would be expected, the resistance generally increases as the amplitude of vibration increases. About one crystal unit in eight, however, exhibits a steady decrease in re- sistance as the drive increases. The initial resist- ance of such a unit is usually higher than the average. Note in figure 1-116 that a number of the curves have relatively sharp negative slopes at very low power levels. This characteristic is not uncommon, particularly in the case of harmonic- mode elements, where it has become a problem requiring special test procedures. Harmonic-mode POWER IN MILLIWATTS Figure 1-7 18. Average frequeney deviation versus drive. Bach curve represents the average of a random sample of severai simiiarly constructed units of one manufacturer. Curves having the same letter represent the character- istics of crystal units of the same manufacturer WADC TR 56-156 112 crystel units are now required to pass performance tests at two drive levels. The first is at the normal maximum recommended drive level; the second is to ensure that the resistance falls within specifications when the drive is at a minimum. In the jcase of fundamental thickness-shear elements, sharp negative slopes of the resistance-drive curves at low drive levels are not as common an occur- rence percentage-wise as is suggested by the 9-mc saipples in figure 1-116. Much more likely to be found are resistance curves with the slopes slightly SecKon I Crystal Oscillators more positive at very low drive levels. 1-262. In the design of an oscillator for military equipment a principal consideration is to ensure that the crystal drive does not exceed the recom- mended maximum when one crystal unit of the same standard type, but of perhaps a greatly different resistance, replaces another. If the drive is not kept to the lowest practicable level, the re- sistance of a borderline crystal may well be in- creased beyond the permissible limits, thereby excessively degrading the oscillator stability. CRYSTAL OSCILLATORS For a comprehensive cross-index of crystal- oscillator subjects, see end of Section /. FUNDAMENTAL PRINCIPLES OF OSCILLATORS 1-263. An oscillator can be defined as any physical system having a periodic motion. If the motion is plotted as a function of time, a wave shape or a sequence of wave shapes that fairly accurately repeats itself would be considered the fundamental cycle of a stable oscillator. On the other hand, if there were a continuous change in the wave shape, the oscillator would be classified as being unstable. An oscillator that is unstable in the general sense, may, however, have a stable component of fre- quency, or amplitude, or some combination thereof. Of course, all oscillating systems are unstable to some degree, so that the terms stable and unstable define classifications that are somewhat arbitrary though none the less convenient. 1-264. Oscillators may also be classified according to the way in which the oscillations are controlled. A number of classifications are possible, but of those which consider the oscillating system alone, there are three general types: free, forced, and forced-free. Free oscillators are those whose oscil- lating energy is entirely self-contained in the oscillating state, and whose waveform and fre- quency are determined entirely by the properties of the system. The solar system, purely from the point of view of the planetary motions, is an example. A quartz crystal vibrating freely in space is another. Forced oscillators are those in which the energy, wave shape, and frequency are under the control of an external power source. An ex- ample would be the cone of a loudspeaker, or a quartz filter crystal, where the vibrations are con- trolled by the signal source. “Forced-free” oscil- lators can be described as those which are driven by the energy of an external source, but where the frequency is primarily determined by the prop- erties of the system. Crystal oscillator circuits are of the forced-free type. Again, the classification is somewhat arbitrary, for in the final analysis there are no absolutely free nor absolutely forced oscillations, nor can two systems be rigorously considered as distinct when there is an exchange of energy between them. In fact, fourth and fifth categories are possible. In the one, the frequency control and drive are both inherent in the system, yet not in the same sense as that defined for free oscillators. By a stretch of the imagination, a good example is to be found in the hula dancer. In a fifth category, the energy is supplied by the oscil- lating system, but the frequency is controlled ex- ternally. An example is to be found by considering each limb of the hula dancer as a separate oscil- lating system. Still other categories are possible. Insofar as a crystal oscillator circuit is concerned, as distinct from its power source, we can consider it an independent controlling system in respect to the frequency, but only to the extent that the cir- cuit can predetermine the periodic characteristics of both the input and the output energies. FUNDAMENTAL REQUIREMENTS OF STABLE FORCED-FREE OSCILLATIONS 1-265. There are two fundamental conditions that are always met when a physical system is being maintained in a stable state of forced-free oscil- lation. First, the primary source of energy, or “prime mover,” is supplying energy at the same average rate at which energy is being expended WADC TR 56-156 113 Section I Crystal Oscillators by the system. Second, all forces acting on the oscillations are, themselves, stable periodic func- tions which have frequencies relative to the fre- quency of the oscillator that can be expressed by f 18 rational numbers (e.g., = 1 . 2 » 7 ’ 1 *' practical case, this latter property cannot occur simply by coincidence between independent sys- tems, so that the condition implies that the periods of all forces acting on the stable oscillations are controlled by the oscillator, itself. The most im- portant of these forces are those exerted by the power source in driving the oscillator and those exerted by the output as a result of reaction with the load. The first condition for stable oscillations ensures that the average amplitude of oscillation is stable. The second condition ensures that the fundamental frequency is stable. Together, they ensure that the waveform is stable. 1-266. As a simple example of a forced-free oscil- lator, consider a system consisting of a swinging pendulum. If the frictional losses per cycle are small relative to the energy stored in the system, and if the amplitude is small, the oscillations are essentially those of simple harmonic motion, with the frequency being determined by the geometry of the system and the gravitational field. Assume that the pendulum, each time it reaches a certain point in its cycle, triggers a latch that releases a spurt of energy from a power source. If each kick received imparts the same amount of energy to the system, an equilibrium of stable oscillations will be reached when the input pulses are being com- pletely transformed into a simple harmonic flow of energy to the surroundings. In order for the power source to transmit energy to the system, it must exert its force while the system is moving in the direction of the force. Otherwise, it will be the “power source,” rather than the oscillator, that gains energy. If the system dissipates energy at the exact instantaneous rate at which it is received, the applied force will not, itself, produce momen- tary accelerations in the pendulum’s swing each cycle. However, the losses from the system do not occur at simply a single interval during the cycle, but obey a sine-wave function extending over the entire period. Only at the instants of zero kinetic energy, at the end of each swing, can the instanta- neous losses be considered zero. Thus, each pulse of energy must accelerate the pendulum in its di- rection of motion so that the waveform must deviate somewhat from a pure sine shape. The dis- tortion is a minimum when the ratio of the total stored energy to the energy input per cycle is a maximum, and when, during the input interval, the ratio of the energy dissipated to the energy absorbed is a maximum. In other words, the dis- tortion becomes smaller the higher the “Q” of the pendulum, the longer the interval over which the impulse is spread, and the more the impulse is centered at the middle of the swing where the in- stantaneous power dissipation is the greatest. Now, even though the impulses distort the wave- form, the oscillations are stable if exactly the same pattern is repeated periodically. If instead of swinging back and forth, the pendulum is swing- ing through a complete circle, it is easier to see that if several impulses are transmitted to the system each cycle, the same pattern will continue to be repeated as long as the frequency, or fre- quencies, of the impulses are related to the frequency of the pendulum by a ratio of whole numbers. The fundamental of the pendulum cycle need not equal the fundamental of the stable wave, but it must be a harmonic thereof. For example, suppose that a pulse of energy is imparted to the system only once every four cycles. Then the funda- mental period of the stable waveform is four times the period of the pendulum. If five impulses are delivered for each four cycles of the system before being repeated in the same phase as before, the period of the stable waveform will again equal four pendulum cycles. Only when there are one, two, three, etc, impulses repeated each cycle does the period of the stable wave equal the natural period of the oscillator. In the same way by which the forces exerted by the energy sources distort the sine-waveform, so do the forces exerted by the load into which the pendulum loses its energy. If the impedance during any interval of the swing changes from cycle to cycle without repetition, a stable waveform is not obtainable. APPLICATION OF FUNDAMENTAL OSCILLATOR PRINCIPLES IN THE DESIGN OF ELECTRONIC OSCILLATORS 1-267. In the application of electronic oscillators, it is not usually a stable over-all waveform that is the first requirement, but a stable fundamental frequency. Practically, however, these two effects are not independent, and the generation of the one involves the generation of the other. *1116 deviation from a pure sine wave in the a-c output of a stable crystal oscillator will be entirely due to the pres- ence of harmonics of the fundamental. Such frac- 5 tional components as -r ths of the fundamental 4 or the harmonics thereof do not appear. With the WADC TR 56-156 114 load impedance constant, the conditions of stability are reached when energy is being supplied at the average rate of dissipation and at the same phase interval of each cycle. In the generalized crystal oscillator circuit of figure 1-108 (B), the first con- dition is met when p, is equal to — R,. The second condition is met when X, is equal to —X,. In ap- plication, the two conditions are not independent of each other, for the build up in the energy of oscillation depends upon the power source not exerting its force in phase opposition to the oscil- lations. Indeed a common approach to the analysis of an oscillator circuit is to establish a single equa- tion that expresses simultaneously the equilibrium requirements of both the rate and the phase of the energy supply. Equation 1 — 289 (1) for the Pierce and Miller circuits is an example. Since this type of equation, when fully developed, usually becomes quite cumbersome, such an approach is only oc- casionally followed in this manual, it being more convenient to treat the two basic equilibrium con- ditions separately. The first condition, that the rates of energy supply and of energy dissipation be equal, can be assumed to be satisfied if the mag- nitude of the rms voltage between any two points in the circuit is constant. This condition can be expressed by an equation which equates the loop gain to unity. By this we mean that, starting with the input circuit, or at any convenient point, the overall voltage gain around the oscillator loop back to the starting point is unity at equilibrium. If the loop gain is greater than unity, oscillations build up, if less than unity, they do not start, or, if already started, they die down. As an example, the loop-gain equation of a simple oscillator of the type shown in figure 1-177 (D), where the feedback energy is transformer-coupled from the plate cir- cuit to the grid circuit, can be expressed as follows ; G,G.G3 =1^^ • • |£. = 1 1—267 (1) where G, is the gain of the vacuum tube, Gz is the gain of the transformer in the plate circuit, and G:t is the gain of the feedback from the transformer secondary to the vacuum-tube input. Although the loop-gain equation may at first glance appear trivial since the product of the voltage ratios equals unity regardless of what voltage values are assigned, it should be remembered that a necessary qualitative implication requires that each voltage ratio represent the gain of an actual transfer of energy from one circuit to another. When the voltage ratios are expressed in terms of the circuit parameters an overall network formula is estab- lished that will serve to discipline the oscillator Section I Crystal Oscillators design. In a similar manner, the second condition of equilibrium can be expressed as an equation of the loop phase rotation, in which the total phase shift in the voltage around an oscillator loop is equal to zero, or to some integral multiple of 360 degnrees. Continuing the example of the trans- former-coupled oscillator above : + 6.P -f Ogs = 0 (or 360°) 1—267 (2) where is the phase of E,, with respect to Eg, is the phase of E„ with respect to E,„ and is the phase of Eg with respect to E,. In most cases ap- proximately ideal conditions can be assumed so that the loop phase requirements need only be analyzed qualitatively. For instance, in the ex- ample given of the simple transformer-coupled oscillator, let it be imagined that the plate circuit is to be designed so that the vacuum tube faces a purely resistive load. Thus, 6,,^ will represent the 180-degree phase reversal introduced by the vac- uum tube, fl„|, will represent a counter 180-degree reversal by the plate transformer, so that the principal phase consideration is to design a grid circuit that will allow Eg to be in phase with E. at the desired frequency. The loop-phase considera- tions are much more involved in the case of the conventional one-tube resonator circuit shown in figure 1-119. The loop phase rotation in this gen- eral type of circuit applies to such oscillators as the tuned-grid-tuned-plate, the Hartley, the Col- pitts, the Pierce, and the Miller. It is discussed in detail in following paragraphs. The loop equations are the guides by which the design engineer ap- proaches the basic oscillator problems of obtaining the desired amplitude, the desired frequency, the desired amplitude stability, and the desired fre- quency stability. As a general rule, these four fun- damental design considerations are handled with the aid of the loop equations in the following ways : a. An oscillator is designed to provide a certain amplitude of oscillation by ensuring that the parameters that vary with the amplitude (usually the vacuum-tube parameters) reach their limiting values, as defined by the loop-gain equation, when the desired amplitude is reached. b. An oscillator is designed to oscillate at a given frequency by ensuring that the loop-phase equation holds at, and usually, only at, the desired frequency. Should the loop-phase equation also have a solution at some other frequency (e.g. the loop phase of the transformer-coupled oscillator mentioned above may well equal zero at more than one mode of the crystal’s vibration), the design must ensure that the loop gain is less than unity at the unwanted frequency. WADC TR 56-156 115 Section I Crystal Oscillators c. The amplitude stability is improved by coun- teracting or minimizing variations in those circuit parameters which, as indicated by the loop-gain equation, are most likely to cause changes in the amplitude. d. The frequency stability is improved by coun- teracting or minimizing variations in those circuit parameters which, as indicated by the loop-phase equation, are most likely to cause changes in the frequency. Before proceeding to a discussion of the particular types of oscillators, let us first examine in detail the phase relations of the conventional resonator circuit of figure 1-119 (B). If a firm qualitative understanding of the operation of this type of cir- cuit is had, the reader should be greatly aided in interpreting the physical meaning of equations later to be derived. PHASE ROTATION IN VACUUM-TUBE OSCILLATORS 1-268. The conventional equivalent circuit of a vacuum-tube amplifier is shown in figure 1-119 (A). The equivalent generator voltage is equal to — where n is the amplification factor of the tube, and Eg is the excitation voltage on the grid. Rp is the plate resistance of the tube, and Zi, is the a-c load impedance. The minus sign of the generator voltage indicates a 180-degree phase difference between the equivalent emf and Eg. For oscillations to build up, energy must be fed back in the proper phase from the plate circuit, or from some circuit in a following stage. The control grid of the vacuum tube is effectively an escapement device for controlling the release of energy from the power source. Since this energy must be re- leased each cycle so as not to be in phase opposition to the oscillator, the grid voltage alternations must be “timed” by the activity in the rest of the circuit. This means that a sufficient and properly phased part of the energy released by the action of the grid must be fed back from the plate circuit each cycle, or from some other circuit of a following stage, in order to continue the periodic release of energy. The initial rush of plate current is to be sufficient to shock the circuit into oscillation, and the initial alternating voltage fed back to the grid circuit must be sufficient for the vacuum tube to generate more a-c energy than is lost during the first cycle. Rp increases with the amplitude of oscillations until equilibrium is reached. 1-269. The phase relation between the grid and plate voltages of an oscillator vacuum tube at equi- librium is the same as that which would occur if the grid were excited at the same frequency from an external a-c source and the tube were connected as a conventional amplifier, operating into the same equivalent load impedance it faces as an oscillator. However, only a certain impedance relationship among the components of a particular oscillator circuit can provide a feed-back producing the proper input phase. It is this necessary impedance relationship that determines the frequency. In the usual single-tube oscillator, the equivalent vacuum- tube generator, of voltage — /»Eg, must drive a plate-coupled feed-back circuit that causes the volt- age appearing across the grid to be rotated 180 degrees ahead of or behind the generator emf. The simplest method of reversing the phase is by trans- former coupling. On the other hand, if two tubes are used, the reversal can be accomplished by the second tube alone. Either of these methods can figure 1-119. (A) Equivalent circuit of vacuum-tube amplifier. (B) Equivalent circuit of crystal oscillators of the Pierce and Miller types WADC TR 56-156 116 enable a crystal oscillator to work into a more or less resistive load, so that fluctuations in the circuit parameters can have little effect on the feed-back phase, and, hence, upon the frequency. In the con- ventional parallel-resonant circuits, such as are illustrated in figure 1-109, the phase is rotated as shown in figure 1-119 (C). First, assume an ideal case in which the resistive losses in the feed-back arm are zero. In this case, but only in this case, Zl would need to be resistive. The frequency would be that at which the plate and feed-back arms operate as a parallel-resonant tank. There would be no phase shift in the voltage across Zl, and Ep would be of the same sign as — /lE,. Zp and Z„ the impedances of the plate circuit from plate to cath- ode and of the grid circuit from grid to cathode, respectively, are reactive, and must always have the same sign. Zpg, the plate-to-grid impedance is the dominant impedance in the feed-back circuit, and is always opposite in sign to Zp and Zg. In the ideal circuit, if Zp and Z, are positive, Zp, is nega- tive, so the current, I,, leads Ep, and therefore —nEg, by 90 degrees. If Zp and Z, are negative, Zp, is positive, so I, lags Ep 90 degrees. The voltage across Zp,, of course, would be in phase with Ep in both instances. Since Z, is opposite in sign to Zp„ Eg thus is opposite in sign to Ep, and the required reversal takes place. Note that I, is first rotated in phase with respect to Ep; next E, is rotated in the same direction with respect to I,. 1-270. In an actual circuit, the feed-back losses cannot be zero, so that a 180-degree reversal can- not be obtained in the conventional feed-back cir- cuit alone. This means that Ep must first be rotated by an amount exactly sufficient to make up the difference. Assume first that Rp is much greater than Zl, so that Ip can be assumed to be essentially in phase with the equivalent generator voltage. In Section I Crystal Oscillators this case, an inductive Zl causes Ep to lead the emf, whereas a capacitive Zl causes Ep to lag the emf. Unless the effective Q of the feed-back circuit is very low, Zl, must be very nearly resistive, for the shift in the phase of Ep need not be large. In any event, the rotation of Ep must be in the same direc- tion as that of I, and E,. For this to occur, the susceptance of Zp must be greater in magnitude than the susceptance of (Zp, + Z,). That is, the reactive component of the current through Zp must more than cancel the reactive component of I,. The smaller the value of Rp compared with the value of Zl, the more nearly will Zl control the phase of Ip, and the more detuned must the parallel circuit become in order to obtain the necessary rotation of Ep. If practically all the resistance in the feed- back arm is between the grid and the cathode, as is normally the case when E, is developed directly across the crystal unit, Ep must be rotated through a larger angle than otherwise, thereby requiring the parallel circuit to be detuned to a greater de- gree. This is because Ep must be rotated by an amount effectively equal to the sum of two angles. One of the angles is the difference between the actual phase of I, relative to Ep and the ideal phase of ±90 degrees. The second angle is the difference between the actual phase of Z, and its ideal phase of ±90 degrees. If all the resistance is contained in the large impedance Zp,, only the phase de- ficiency of I, is reflected in the phase of Ep. On the other hand, if all the resistance is effectively con- tained in the small impedance Z„ the effect on the phase of Ep by I, is normally small by comparison with the effect due to the grid-to-cathode resist- ance. Expressed in polar form : — IgZg (^i« + ^zg) “ IgZg (®°) where fii, and dzg, which must be equal in magnitude characteristics of eOUIVALEHT-CIRCUIT PARAMETERS i90* PHASE RELATIONS 180«^ — t-90* {VOLTAGE AND CURRENT WITH RESPECT TO E g. 1 MPE DANCES with respect TO CURRENT THROUGH THEM ) z P ^9 Z pg BO jzp/Zpgcj Zl ip E P 1 g Z g E g -Ht- <OR > ‘ &3 i n ♦ 1# » 1 ♦ ♦ ♦ -nnnp — vw- » 1 » 0.95 / / \ If -onnp' — >> 1 as 0.99 ' — ♦ 1 m •vvvA -TTTinr' — If S3 > < 0.95 / \ II MM ... I# as 1 < 0 95 ♦ * II ( C I Figure I- 1 19. (C) Chart showing ideal and typical phase relations necessary for forced-free oscillations of the circuit shown in (B) WADC TR 56-156 117 Section I Crystal Oscillators but opposite in sign, are the phase angles of I, with respect to Eg and of Z* with respect to I*. Now, letting dzvsc equal the phase of the total feed-back impedance, (Zpg + Z,) , with respect to the current through it, and 5 ep equal to the phase of Ep with respect to Eg, we have ^Kg = = 0 and ^Kp ^ig ^Zpgc so that = Ozpgc — 0z^ 1 — 270 (1) Since flzpg,. is opposite in sign to 0zg, these two phase angles add numerically. If it is assumed that Zpgc is approximately six times the magnitude of Zg, but that all the feedback-arm resistance is between the grid and cathode, the Q of Zpgc will be approxi- mately five times the Q of Zg. Under these condi- tions the rotation of Ep from the ideal value of —180 degrees is approximately 20 per cent greater than the deviation of —Bzg from its ideal value of 3:90 degrees. If Zg represents the effective imped- ance of a crystal unit in parallel with the grid-to- cathode capacitance and resistance, the minimum rotation of Sep occurs when the effective Q of the crystal and its shunt impedance is a maximum, provided Zpg > > Zg. Similarly, if Zpg represents the Zc of a crystal unit whose effective resistance is much greater than the equivalent series resistance of the grid-to-cathode impedance, the rotation of Sep depends primarily upon the crystal unit Q., and is a minimum when Q, is a maximum. 1-271. In a conventional parallel-resonant crystal oscillator having an ideal feed-back arm, the fre- quency would be determined entirely by the reson- ance of the tank circuit, so that fluctuations in Rp, although effective in changing the activity, would not affect the frequency. In practice, the equivalent resistance of the crystal unit is a parameter of the feed-back arm, so that the detuning of the tank becomes very nearly a direct function of the effec- tive Q of the crystal and its shunt capacitance. Note that the phase of Ep with respect to E* and — /lEg is determined entirely by the parameters of the feed-back arm. As long as oscillations continue, variations in Rp or Zp can only change the phase of Ep indirectly, i.e., by causing a change in the Q of the feed-back arm. Since the effective resistance can vary by as much as a factor of 10 between minimum and maximum values for the same stand- ard type of crystal unit, an oscillator cannot be designed too closely upon the assumption that the load impedance Z,. will be essentially the same either in phase or magnitude when one crystal unit is replaced by another — even if it is of the same type and nominal frequency. This limitation might possibly be minimized by switching the crystal unit from the feed-back circuit to the plate circuit, and replacing it with a high-Q inductor in the feed-back arm. If the grid were operated with bias sufficient to prevent the flow of grid current, a very high and predictable Q could be obtained in the feed-back circuit, and an approximately resistive Zi, could be assumed for the tank regardless of the variations in Zp due to variations from one crystal unit to the next. 1-272. An oscillator that can be represented by the equivalent circuit shown in figure 1-119 will show the following phase characteristics and related effects. a. The phase of Ep is entirely determined by the over-all Q of the feed-back circuit and the Q of the grid-to-cathode impedance, Zg. The phase angle, Sep, is given by equation 1 — 270 (1) as being equal to (tfzpgc —0zg)- Let us now assume that (flzi«c —6ze) is determined by an imaginary over-all Q of the feed-back circuit. This we define to be Qr = I cot Sep 1 = I cot (0zpgc - 6zg) \ ^ tan ^zpgc tan dzg 4- 1 tan dzpgc - tan dzg _ Qpkc Qk ~ 1 _ Qi'gc Qk 1—272 Qp« + Qg Qpgc + Qg ^ ’ It can be seen that if either Qpg,., the actual effec- tive over-all Q of the feed-back circuit, or if Qg, the Q of the grid-to-cathode impedance, is very large compared with the other, Qf is approximately equal to the smaller Q. b. For a given phase difference between Ep and — /lEg, the ratio of Zp to the total feed-back imped- ance, Zpgc, is less than 1 by an amount which in- creases as the ratio of Rp/Zt decreases. In other words, the ratio of the r-f current in the plate cir- cuit to the r-f current in the grid circuit increases as Rp decreases. c. The value of Rp/Zi, is partly a function of the 2 ratio = Ep/Eg. Assume, for example, that Rp > > Ze, so that Ep/^Eg (= |^. Also assume that during oscillations Zg is decreased, but that Zpg remains essentially constant. The ratio Ep/Eg is thus increased, and likewise the ratio 2 Ep//iEg In other words, as Eg becomes a Kp smaller component of the total voltage across the WAOC TR 56-156 118 feed-back circuit, Ep cannot decrease in the same proportion, else each succeeding cycle would be weaker than the one before; so the ratio Rp/Zl must decrease. Part of the change is due to the increase in Z,., and part is due to a decrease in Rp. If it is assumed that a large percentage change in Z„ causes only a small percentage change in Zp*,., then Zl remains essentially constant in magnitude and Rp becomes the principal variable. In any event, as Rp/Zj, decreases, the effective Q of Zl, as represented by an equivalent resistance and react- ance in series, must increase in order to compen- sate for the increased phase shift of Ip. 1-273. From the qualitative discussion in the fore- going paragraphs it can be seen that in the con- ventional parallel-resonant crystal oscillators the state of an oscillator in operation is primarily de- termined by the impedance relations in the feed- back arm. Since the impedance of a crystal unit changes very rapidly with a small change in fre- quency, a crystal connected in the feed-back circuit makes the oscillator less critical in design than would otherwise be the case. Where maximum sta- bility is required, the vacuum tube will be operated as nearly class B as possible. Under class-A con- ditions, Rp and n are approximately given by the d-c, plate-characteristic data of the tube. In the case of power oscillators, amplifier operation will normally be class C, although class B or even class AB may be employed in particular circuits. In these cases, the effective tube parameters cannot be known beforehand, but reasonably accurate ap- proximations can be made and optimum operating conditions can be reached by more or less trial- and-error final adjustments. The operation of con- ventional oscillators is made less critical, both in starting oscillations and in maintaining a constant amplitude, by the use of gridleak rather than fixed bias. From the point of view of phase rotation, the conductance of the gridleak somewhat decreases the Q of Zb, and thereby necessitates increased detuning of the tank. Nevertheless, in low-power oscillations the gridleak losses can normally be considered negligible in comparison with the crys- tal losses. It is as a limiter and stabilizer of the amplitude that the gridleak bias is most important. Any changes in the circuit that tend to change E, automatically change the bias in such a direction that Rp and g„, of the tube are readily adjusted to new equilibrium values, so that the tendency is one of immediate opposition to the change in so far as the activity is concerned. TYPES OF CRYSTAL OSCILLATORS 1-274. Crystal oscillators are frequently classified WADC TR 56-156 Section I Crystal Oscillators as being either crystal-controlled or crystal-stabi- lized. A crystal-controlled oscillator is defined as an oscillator that cannot oscillate if the crystal is removed or is defective. A crystal-stabilized oscil- lator, on the other hand, operates as a “free-run- ning” oscillator if the crystal is removed. When the crystal unit is properly inserted and the “free- running” frequency is made to approach the nor- mal resonance of the crystal, the mechanical vibrations of the crystal sharply increase. At some point the piezoelectric effect will be sufficient to suddenly “capture” the oscillations and thereby synchronize them at the crystal-circuit frequency. Generally, the crystal-controlled oscillator is pre- ferred, since it is not desirable that oscillations continue if the crystal unit suddenly or gradually becomes defective. Also, crystal-controlled oscilla- tors are normally less critical to design and are less likely to jump suddenly from one frequency to another. The crystal-stabilized oscillator does have the possible advantage of being able to oper- ate successfully with very-high-Q crystal units whose piezoelectric coupling, however, would be too weak for the crystal to build up oscillations from a single initial impulse. A number of oscil- lator circuits appear to be border-line cases, that can only arbitrarily be classified as crystal-con- trolled or crystal-stabilized. For example, the Cl meter circuit shown in figure 1-106, if connected in the crystal position of Si, would fail to oscillate if the crystal terminals w'ere open, but not if they were shorted. Most of the oscillator circuits that are discussed in the following paragraphs are classified as crystal-controlled inasmuch as the oscillations do not occur if the crystal units are disconnected. 1-275. A more practical classification from the standpoint of circuit design and of selection of a crystal unit is that of series- and parallel-resonant crystal oscillators. In general, the series-resonant type provides the greater frequency stability and can generate the higher frequencies; whereas the parallel-resonant type is the more economical to construct, can operate over a wider frequency range by the substitution of different crystal units, and can generate the greater power output. There are, nevertheless, a number of exceptions to the general rule. PARALLEL-RESONANT CRYSTAL OSCILLATORS 1-276. The first quartz oscillators to find general usage as frequency-control devices were of the parallel-resonant type. These oscillators are used primarily with fundamental-mode crystals at fre- 119 Section i Crystal Oscillators quencies below 20 me. In the conventional circuits of this type, the crystal must operate between its resonant and antiresonant frequencies, thereby be- having as an inductor. Under these conditions the circuit does not oscillate if the crystal unit becomes defective. Unless a frequency monitor is to be available, these oscillators should be permitted a tolerance of 0.002 per cent or greater, depending primarily upon the tolerance rating of the crystal unit to be used. Under extreme operating condi- tions, an oscillator error of approximately twice the crystal unit tolerance should be permitted. The conventional circuits are of the Pierce and Miller types, or their modifications. Maximum stability is achieved with low crystal drive and class-A to class-B operation of the tube. Maximum power effi- ciency is achieved with class-C operation. Since fundamental-mode crystals become too thin and fragile for operation above 20 me, overtone crys- tals are necessary at these higher frequencies. However, the shunt capacitance, C„, and the series- arm L remain the same whether the thickness- shear crystal is operated at the fundamental or the overtone mode, whereas the series-arm C varies inversely with the square of the harmonic. Thus, the capacitance ratio, Co/C, increases with the square of the harmonic, so that the electromechan- ical coupling may be too weak to initiate oscilla- tions at normal voltages if the crystal unit is to be operated at parallel resonance. For this reason and also because Co, as well as the tube capaci- tances shunting the crystal, have larger suscep- tances at . the higher frequencies, which greatly reduce the operating range of the crystal unit, the parallel-mode circuits are unsuitable for use at the higher frequencies. If the basic circuits are modi- fied to employ series-mode crystals, or are used in conjunction with frequency-multiplying stages, they can provide stable control of very high fre- quencies. The introduction of a number of multi- plier stages with the attendant problems of pre- venting unwanted frequencies is usually less to be preferred than the direct generation of the end frequency by the use of overtone-mode crystals in series-resonant circuits. Although frequency multiplication involving more than one multiplier stage is still widely used in conjunction with par- allel-resonant master oscillators, this usage is found principally in medium- to high-frequency transmitters where various multiplying combina- tions can provide the maximum number of chan- nels with a minimum number of crystal units. Control by parallel-mode circuits of frequencies above 30 me is not very common. One example is to be found in Radio Set AN/ARC-IA. In this equipment, a fundamental frequency of 8 me, for instance, would be doubled in the oscillator plate circuit and increased nine times more in the plate circuit of the following stage. Thus, with only two tubes, a parallel-resonant oscillator can control a frequency of 144 me and higher. In the analysis of the particular oscillators to follow, the Pierce oscillator has been chosen as something of a refer- ence circuit as well as a point of departure in the discussion of many design considerations to be encountered in crystal oscillators. For this reason, the reader will find the treatment of the Pierce circuit, both qualitatively and mathematically, con- siderably more detailed than that of the other types of circuits. Space forbids as comprehensive a treatment for the other circuits, but the design problems and methods illustrated in the particular case of the Pierce oscillator are applicable in prin- ciple to all oscillatoi s. The Pierce Oscillator 1-277. The Pierce oscillator is fundamentally a Colpitts oscillator in which the plate-to-grid tanl^ inductance has been replaced by a crystal unit, as shown in figure 1-120. The design of the Pierce oscillator is simpler and less critical than that of any other crystal circuit. As long as Ri and R* are large compared with the capacitive reactances shunting them, the Pierce circuit will oscillate with crystal units covering a wide band of frequencies. The use of load resistance, Ri, in figure 1-120 (A) aids in maintaining a reasonably flat response over a wide frequency range without the necessity of tuning adjustments other than the switching from one crystal unit to the next. Where a broad fre- quency range is not required, or where greater activity is necessary, an r-f choke should be used in place of Ri; otherwise, power approximately equal to I,,'‘Ri is simply wasted (Ib = average d-c plate current). But even with this economy, the Pierce oscillator cannot be used to generate as large an output as the Miller circuit. The principal reason is to be found in the fact that the imped- ance Z,,g of the equivalent circuit, which in this case is provided by the crystal unit in parallel with the plate-to-grid capacitance of the tube, must be approximately equal to, or greater than, Zp and Zg combined. Since the impedance of the crystal unit is fixed by its frequency and rated load capaci- tance, larger plate impedances are possible if the specified crystal impedance is Zg instead of Zpg. Thus, for the same crystal current, larger output voltages can be developed across the tank in the Miller than in the Pierce circuit. On the other hand, the effective feed-back phase Qf is greater WADC TR 56-156 120 S«cHon I Cryital Oscillators if the crystal is not connected between the grid and cathode. This permits the tank to appear more nearly resistive to the tube, so that fluctuations in Rp have less influence upon the frequency. Thus, a Pierce oscillator is generally more frequency stable than a Miller oscillator using the same crys- tal unit. The typical Pierce circuit employs a triode, although screen-grid tubes are usually to be pre- ferred, because the higher Rp and the negligible plate-to-grid capacitance serve to improve the fre- quency stability. The oscillator is generally used at frequencies above 200 kc and below 15,000 kc. If an overtone mode is to be excited the oscillator must be made frequency-selective, by replacing Ri with an inductor, Lj. The inductance of Lj must be such that the antiresonant frequency of Li in par- allel with Cl is lower than the operating frequency of the crystal, in order that Zp will appear capaci- tive. The use of the inductor-capacitor combination also reduces the harmonic content of the output waveform. If small L/C ratios are used, the effec- tive plate-to-cathode capacitance will be much greater for the overtones than for the fundamental frequency, so that the former are more readily bypassed to ground than would be the case if no coil were used. In deciding upon the type of oscil- lator circuit to use, those rule-of-thumb factors most favorable to the selection of a Pierce cir- cuit are : a. The frequency lies between 200 and 15,000 kc. b. The permitted frequency error is not less than 0.02 per cent, or 0.015 per cent if a regulated voltage supply is available. c. The oscillator is to be capable of untuned operation over a wide frequency range, simply by switching from one crystal unit to another. d. Only a small voltage output is required. e. The oscillator must be of inexpensive design. f. The oscillator must not be critical in opera- tion, but able to oscillate readily with relatively large deviations in the parameters of the external circuit. g. Wave shape is not critical. h. Same as above, except that the permitted frequency error is 0.01 per cent and thermostatic control of the crystal temperature is feasible. i. Same as above, except that the permitted fre- quency error is 0.005 per cent, thermostatic con- trol of the temperature is feasible, a regulated figure I-I20. Diagrams illustrating the equivalence (B) between the Pierce circuit and the Colpitts circuit colpitts oscillator WADC TR 56-156 121 Section I Crystal Oscillators ANALYSIS OF LOAD CAPACITANCE, C., IN PIERCE CIRCUIT 1-278. Once that it has been decided to employ a Pierce type oscillator, the standard type of crystal unit is chosen which provides the desired fre- quency and frequency tolerance and which has been tested according to the Military Standards for parallel-resonance operation. One of the first design considerations is to ensure that the crystal unit will effectively operate into its rated load capacitance, C,. Such operation is necessary, else there can be no assurance that one crystal unit of the same type can replace another and still fall within the drive-level and effective-resistance spe- cifications. For most parallel-resonance crystal units the value of Ci is 32^/if, although at frequen- cies under 500 kc, values of 20 /i/if are common. In particular instances, still other values of Cx are designated. To a first approximation, referring to figure 1-121 (A), the crystal unit operates into a load capacitance equal to Cpg plus the total of Cg in series with the parallel combination of Ci, Cpc, and the effective inductive impedance presented by the vacuum tube. Since the Q of the feed-back arm is not infinite, Ep, it will be recalled, must be rotated slightly away from — /^Egi the direction is such that for a particular frequency Cpg -f Ci must be slightly larger than would otherwise be the case. Even though the actual equivalent tank circuit is slightly detuned, mathematically the crystal unit is to be in resonance with an effective load capaci- tance Cg. (See figure 1-108 (D).) The vacuum tube appears to the tank circuit as a negative re- sistance having a positive reactive component sufficient to cancel the excess susceptance of Zp. At equilibrium, the tube can be represented by an equivalent inductance, Lr, in parallel with & nega- tive resistance, pr, as in figure 1-121 (A). Note that pT is smaller than p of figures 1-121 (C) and (D). This is because pT is not connected directly across the crystal, but faces an impedance, ap- proximately Zl, that is less than the crystal PI. In figure 1-121 (B), Lt has been replaced by an equivalent negative capacitance, Cp. If Cpg can be considered negligible, X/ and R/ are equal to Xg and Rg, the equivalent parameters of the crystal unit alone ; otherwise, the values of Xg' and Rg' are based upon the assumption that the shunt capaci- tance, Co, of the crystal has been increased by an amount equal to Cpg. In figure 1-121 (C), Cp (= Cpg H- Cl) and C„ are shown combined into a single plate-to-cathode capacitance, Cp'. In figure 1-121 (D) , Cp' and Cg are represented by a single load capacitance, C,'. If Cpg is negligible, C,' be- comes the equivalent load capacitance, C„ into which the crystal unit operates, and it should be i.m. PI.-. X.- ..j p...™ -k. !»»». .p.. -k. r if Ik. r .3....- »fc.. positive reactance equal to that of Lt- h « me r t piaio surr* WADC TR 56-156 122 equal to the value specified for the particular crys- tal. In any event (Ci' + Cp*) is the load capaci- tance that the crystal unit faces, and which should be equal to the rated value, C,. The value of C, includes not only the tuning capacitance in the plate circuit, but also the distributed capacitance of the output leads as well as the effective input capacitance of the next tube. C* is the input capacitance of the oscillator tube. The losses due to R„ the effective grid-to-cathode resistance, and to Rl. the effective load resistance, have been as- sumed to be negligible compared with those in the crystal unit. These assumptions can be made with- out appreciable error in a low-power Pierce cir- cuit that requires only a minimum of loading. From an inspection of figure 1-121 it can be seen that in order for a crystal unit to operate into its rated load capacitance, the design must be such that c. = Cp* -f C,' = Cp, -h 1—278 (1) Effect of Cpg in Pierce Circuit 1-279. Cpg effectively increases both the reactance, Xp, and the resistance, R„ of the crystal unit. This does not mean that the true effective Xe and Re are changed, for these are fixed by the fact that X* must resonate with the rated load capacitance, Cj. Still, insofar as the impedance from plate-to-grid is concerned, Xe and Re effectively have increased values which make it appear that the crystal shunt capacitance is increased by an amount equal to Cpg. This effect is not desirable, since Re is effec- tively increased by a greater percentage than Xp. (See’ equations (1) and (2) in figure 1-98 for the effect on Xe and Re if X, is held constant but Xco is decreased.) Thus, the larger the value of Cpg, the smaller the Q of the feed-back arm becomes the more the tank circuit must be detuned, the greater must be the value of the negative capaci- tance, C„, and hence the greater the frequency in- stability due to changes in the tube parameters. The plate-to-grid capacitance needs to be consid- ered only if a triode is used or if the crystal unit is oven-mounted. In the average triode Cpg is on the order of 1.5 to 2.5 fi/it, sufficient to increase the effective value of C„ by as much as 50 per cent in some cases. When the second grid of a tube is used as the oscillator anode, as in the case of penta- grid converters, Cpg is usually on the order of 1 n^tf. The pin-to-pin capacitance introduced by ovens may be as high as 5 /i/if. The Cpg of screen-grid Section I Crystal Oscillators tubes can all but be neglected, since the increase in capacitance across the crystal is only about one- thousandth of the total. Because of its negligible Cpg, a pentode is preferred when the frequency deviation must be kept to a minimum. In the re- maining discussion of the Pierce circuit, we shall assume that Cp* is negligible, so that X/ and Re' will represent the actual effective impedances of the crystal unit, and C/ will equal the rated capaci- tance, C,. Although we shall employ the unprimed symbols X, and Re to designate the plate-to-grid impedances, it should be remembered that this is only a convenience, for where X, is predetermined by the rated load capacitance and the frequency, Xe' necessarily increases or decreases, respec- tively, with increases and decreases in Cpg. Simi- larly, Cx' varies negatively with Cpg, but we shall assume that it is a constant equal to Cj. Determination of the Effective Negative Capacitance, C„, Introduced by Vacuum Tube in Pierce Circuit 1-280. First, in order to avoid possible confusion, it should be pointed out that the selected reference or zero phase angle of the equivalent circuit in figure 1-119 is not the same as that implicitly as- sumed in the negative-resistance circuit of figure 1-121 (A). In figure 1-119, the reference phase has been taken as the phase of Eg, whereas in fig- ure 1-121 (A) it is the phase of the current through the negative resistance pr (not Ip), which in turn is the same as the phase of the r-f plate voltage, Ep. Now, Ip, in the negative-resistance circuit, is physically the same as the Ip of the vacuum-tube generator circuit. The equivalent current through p-r represents that component of Ip in phase with Ep — not that part of Ip in phase with — /iEg. The current through the negative re- sistance is thus smaller in magnitude than the total r-f plate current. The imaginary current through Li-, or C„, is equal and opposite to that component of Ip which is 90 degrees out of phase with Ep. In the phasor chart in figure 1-119 (C), the bottom line shows the phase relations that are approached in a Pierce circuit if Rp of the tube approaches Zl in magnitude. Note particularly that Ep and Ip rotate in opposite directions. Ep must lag Ip by an angle whose tangent is at least as great as „ — ; that is, the tangent of the angle cannot be less than the reciprocal of the Qf of the feed-back arm. The minimum angle occurs when Rp is very much greater than the load imped- ance Zl and the gridleak losses are negligible. On the other hand, the phase difference between Ep WADC TR 56-156 123 section I Crystal Oscillators and Ip cannot be greater than 90 degrees, for the simple reason that Ep is a counter emf produced by Ip flowing through an equivalent impedance, Zl, which has no component of negative resistance. There is another limitation in that the rotation of Ip with respect to — /lEg cannot exceed 90 degrees minus the necessary rotation of Ep with respect to — /iEg. Otherwise, the necessary rotation of Ep cannot occur. As this extreme is approached, Zl approaches a pure reactance approximately equal to Xcp, Ip approaches a 90-degree phase lead over Ep, and the apparent Q of the entire plate circuit (approximately Zl/Rp) approaches the Q( of the feed-back circuit. These maximum and minimum phase angles are summarized in the following table. The phase angles are defined by the absolute values of their tangent expressed in terms of the phase Q of the feed-back arm, Q,. Also shown are the limiting Q’s of Zl and (Zl -|- Rp) : which re- spectively determine the phases of Ip with respect to Ep and — /lEg. In the last column are shown the limiting values of C„ which are discussed in the following paragraph. Angle of Ep at all times With Respect to : Q of; Cp (Approx) Ep — ftE, B Rp -|- Zl 0® tan-‘(l/Q,) ^■’(Of'+Rp + R.) Minimum angle of Ip t4n-‘(l/Q,) 0” 1/Qf 0 -Cp/Qf* Maximum angle of Ip O O tan-’(Qf) 00 Qf CpR, Rp + R« 1-281. The approximate expressions given in the table above for the limiting values of C„ are de- rived upon the assumption that the unsigned phase angle between Ip and Ep is given by the equation tan' tan' -1 / Xzi, \ \ Rp Kzp / + where Xzl and Rzl are the absolute values of the equivalent series reactance and resistance whose vector sum is equal to Zl; tan~> (i)" the un- tan-’ signed phase difference between Ep and — and ’ ( p • ) is the unsigned phase difference \ Kp -f* iCzl / in the opposite direction between Ip and — From equation (1) it follows that Xz = tan tan 1—281 (2) On applying the general trigonometric equation for the tangent of the sum of two angles tan (x -I- y) tan X ■+• tan y 1 — tan X tan y Letting tan x = 1/Q( and tan y = Rp + Rj -, equa- 1 281 (1) ^2) becomes Xzl Rzl 1 Xzl Qf Rp -f- Rz 1 - Qt (Rp + Rzl) Rp ~h Rzl ~h Qf Xzl Q( Rp -|- Qt Rzl — Xzi 1—281 (3) On rearranging, equation (3) can be expressed as an equation for Rp ; Rp = Rzl^’ + Xzl" ^ Zl* Qf Xzl — “ Rzl Qf Xzl Rzl 1—281 (4) WADC TR 56-156 124 Figun 1-122. Gen«ralU«d Pierem osci/Zafor. Xz^ and Xcn npresant the potiiiva magnitudes of the equiva- lent reactances of the load and the dynamic effects of the tube, respectively, pr Zfie same as the pr in figure 1-121 Note that for oscillations to be maintained, QjXz,, must be greater than Rzl, else Rp becomes nega- tive. Referring to figure 1-122, the effective ad- mittance of the Zl and Xcn branches in parallel is equal to - -. From this expression Rzl — jXzt jXco it can be shown by straightforward manipulation that if the reactive component of the admittance is to be zero Xc„ Xz, ^ Xz.* = 1-281 (5) On substitution of equation (5) in equation (4) and rearranging Xc. = Rp (q. - Ij) 1-281 (6) The values of C„ as listed at the end of paragraph 1-280 are obtained from equation (6) by substi- tuting the values of Qf and Rzp/Xzp when these are expressed in terms of the basic circuit param- eters. In paragraph 1-289, it is shown that at equilibrium : Xcp + Xc. + X. -f = 0 1—289 (3) Rp The term, accounts for that part of the Kp negative capacitance which is necessary to com- pensate for the phase shift in Ip, but not for that part which compensates for the phase shift of Ep. For example, if Rp were infinite, the phase shift of Ip, would be zero, and likewise the term, Section I Cryctcil Oscillalory in equation 1 — 289 (3). Nevertheless, Ep must still be rotated by an angle equal to tan-'r^^, so that the tank cannot actually be parallel-resonant. Some value must be assigned to the negative ca- pacitance, for the apparent resonance to hold in the generalized negative-resistance circuit. Now, Letting and Zl Zp Zpip; Zp + Zp,c *Zp — jXcp Zpgo = Re + j(Xe -f- Xc*) we can express Zi, as a complex function equal to Rzt — jXzL, where Xzp is still assumed to be un- signed. Thus, (complex) Zl On multiplying both numerator and denominator by ^ Re 4- that Rzl = ^Cp ^Cp R. Re 1—281 (7) and Xzp = Xcp I [ R, + (Rp + R.) ] Xcp I (Re Rp + Xcp^ ) Re Rp 1—281 (8) Xcp I Rp’* So Rzl _ X*L " R,Rp* + Xcp* (Rp + R.) I Xcp I Rp R R 4- X ^ ^-281 (9) •Ke-Kp Acp Also, when assuming that the grid losses are negligible, X. 4- Xc* ^ I Xcp I (Rp 4- Re) R, Re Rp 1—281 (10) Q. = WADC TR 56-156 125 section i Crystal OKillatart The expression on the right in equation (10) is obtained by substitution from equation 1 — 289(3). On substituting equations (9) and (10) in equa- tion (6), we have Xc„ = I Xf I r ^P ~l~ ^P* 1 ' ' L Re R. Rp -!- Xcp* (1 + Re/Rp)J Xc„ = I Xcp I [ RpXcp" (l+Re/Rp)-hReX-t>R.Xcp^ (l-hR./Rp) ~| L Re' Rp + ReXcp' (1 + Re/Rp) J 1—281 (11) Equation (11) is obtained by using the exact values of Rzl/Xzp and Qj as given by equations (9) and (10). Although the equation for Xcn involves the difference between two nearly equal terms, the error introduced by using the approximate values of Qf and Rzi,/Xzl is negligible for all practical purposes as long as Rp >> R,. Now, Cp = 1 wXcn SO, on substitution of equation (11), Cp — “ Cp r Re' Rp -h R. Xcp' (1 + Re/Rp) 1 LRpXcp' (l+Rp/Rp) -I- R.'Rp + RpXcp* (H-R./Rp)J 1—281 (12) In the practical case, Rp is much greater than R, and X cp’ is much greater than R,’, so that the ap- proximate equation for Cp becomes r - r r R.’’ Rp + R. Xcp' 1 ~ Rp (Xcp' -h R.') J The last term on the right inside the parentheses Xrn* is obtained by assuming that ^ < 7 p « is more Aop “H XV* It nearly equal to than to 1. We make this iVp -f- K* assumption arbitrarily for the convenience in re- membering the limiting values of Cp. That part of Cp which is necessary so as to compensate for the phase shift of Ep with respect to — /lE, is ap- proximately equal to — Cp/Qf*; whereas the part necessary to compensate for the phase shift of Ip with respect to — is approximately equal to ^ s — r^rr- R is this latter component that is ac- Kp + K* X R counted for by the term, * , in equation 1 — ^289 Kp (3). By equation (12), when Rp/Zl approaches infinity, the limiting value of C„ is found to be Cp (Rp/Zl -» oo) -CpR,' Xcp' + R.' ^ 1-281(14) Wi When Rp/Zl approaches its smallest possible value, the ratio Rzp/Xzp becomes negligible com- pared with Qt, so that Xcb by equation (6) is ap- proximately equal to QfRp. Substituting equation (10) for Q„ we have, when / — \ Rp -f- Rzl / “ «Q,Rp R. w| Xcp|(Rp -l- R.) — CpR, Rp + R« 1—281 (16) As indicated in figure 1-121, the total effective plate capacitance is 1—281 (16) When Rp > > Zl, the Q of Zl approaches 1/Qt in value and Cp'=Cp(l--^) 1-281(17) When Rp is small compared with XcpVR*. then Rzl/Xzl becomes small compared with Qf and Cp' = 1—281 (18) Rp + R, Rp is normally much greater than R,. Only low- frequency crystals have effective resistance which approach in value the plate resistance of low-power vacuum tubes. For all practical pur- poses in the average Pierce circuit, Cp' can be as- sumed to equal Cp, the static plate capacitance, except when considering problems of frequency stability. What is important to note in the limiting equations for C, is the fact that if the tank is to WADC TR 56-156 126 Saction I Crystal Oscillators be operated well off resonance, Rp becomes quite an important factor in determining the frequency. In this case, because C„ is relatively large, any variation in the tube R„ has a great effect upon the frequency. It should be remembered that the parameter Qt has been used to account for the re- quired rotation of Ep with respect to —/j-Eg. Inso- far as the gridleak is effective in increasing the necessary phase shift, Qt = — ^ cannot be assumed, and the complete equation 1 — ^272 (1) must be used. THE EFFECT OF R, UPON THE FREQUENCY OF PIERCE CIRCUIT 1-282. It can be seen that for large values of the Ri,/Zi, ratio, C„ is small and its percentage varia- tion with changes in Rp is smaller still. Cp is, effec- tively, a frequency-determining parameter, but more exactly it is a mathematical function that in- directly expresses the effect of Rp and Qf upon the frequency. The smaller the Qt and Rp, the larger is C„; and the larger C„, the greater is the effect of Rp. Since Rp is subject to change with changes in the tube voltages, tube aging, and the like, it is important to keep C„ as small as possible. This can be done by designing the circuit to operate with as high of value of tube Rp as is practicable. For a given tube, the higher values of an effective Rp are to be obtained when the tube is conducting during only a small fraction of a cycle. This in turn re- quires that the oscillator tube be operated class C, so that a larger grid bias than otherwise is re- quired. However, if the crystal drive level is to be kept low and if the gridleak is to have a negligible effect on the effective Q(, and hence upon C„, the gridleak resistance must be as large as practicable without running the risk that the tube will block or operate intermittently. The limiting value of Rp occurs when — is just sufficient to maintain oscillations. If the vacuum tube could operate into a pure resistance. Ip would be in phase with — /xE*, and Rp would be eliminated as a frequency-deter- mining element. In the conventional Pierce circuit this could occur only if Qt were infinite. Pkase-Stabilized Pierce Circuit 1-283. If an inductor is inserted in the plate cir- cuit of the oscillator, as indicated in figure 1-123, having a reactance equal and opposite to the effec- tive reactance Xz,., then Ip undergoes no phase rotation, and changes in Rp, although affecting the activity, will have little effect upon the frequency. With Ip in phase with — /xEg, the Q of Zl must equal 1/Q,, and the operation of the tank is the same as it would be if Rp were infinite. If Qf is reasonably large and is approximately equal to IXco/R*!, R*x = Zi, = XcpVRe = ! Xcp IQ. Ip iXu-iXZL Figure I-I23. Coil inserted in plate circuit of Pierce oscillator to prevent phase of ip front being influenced by changes in Rp Since Xzp/ Rzi. = 1/Qf XzL is approximately equal in magnitude to Xcp. Thus, for the vacuum tube to look into a resistive load, the inductor should have a reactance approxi- mately equal in magnitude to X,-,,. This value as- sumes that the gridleak and output losses are negligible. When such assumptions cannot be made, the value of the series plate reactance be- comes a more involved function. Llewellyn ana- lyzed this type of circuit and eliminated Rp from the frequency-determining equation (phase rota- tion equation) by equating the sum of the factors of Rp to zero. Although the approach is different and the grid losses are assumed to be predominant, Llewellyn’s mathematical elimination of the ef- fects of Rp upon the frequency by the introduction of a plate inductor in series with the tank appears to be equivalent to the qualitative condition that I„ must be held in phase with — juE,,. The experi- menter, nevertheless, should be warned that the theory of this type of stabilization has been ana- lyzed above, and also by Llewellyn, only in terms of the phase relations. Difficulty will probably be experienced in obtaining .stable oscillations with- out additional modifications to ensure that the limiting • characteristics are changed from a voltage- to a current-controlled nature. This fea- ture of oscillator theory has not been fully ex- plored, but see paragraphs 1-585 to 1-598 for a general discussion, and paragraph 1-323 for a particular example of an attempt, which was not entirely successfu', to . phase-stabilize a Pierce circuit. Conditions for Maximum Rp in Pierce Circuit 1-284. Referring to figures 1-119 (A) and (B), we shall begin with the assumption that the tank is Z ^ operating near resonance so that Zl«-^— , where R, (not shown) is the effective resistance of the crystal unit whose total impedance is represented by Zl> therefore, is very nearly resistive, and Ir is approximately equal in magnitude to the cur- rent through Zp. Thus, Ig « Ep/Zp WADC TR 56-156 127 Section i Crystal Oscillators also.E, = I.Z. = Rearranging, Ep/E, ^ = C,/Cp In the interest of maxium stability it is desirable for Rp to be a maximum. The problem is to find what capacitance ratio, Cg/C,„ permits the largest possible value of Rp consistent with the rated drive level and load capacitance of the crystal unit. The phase-rotation equations do not enter the problem — only those equations that concern the magnitude of the equilibrium voltages and currents are of concern now. The crystal specifications indirectly set an upper limit for the tank current, I,. Thus, the output voltage, Ep I* Zp, also has an upper limit, since Zp (= Zp* — Zg) has a theoretical maxi- AndL = _ mE, pEpZg Rp + Zl Zp(Rp -h Zl) SoEp = — ^ Zp(Rp -f- Zl) or fiZgZi, — RpZp — ZpZj. On substituting (Rpgm) for n, where gn is the transconductance of the tube. RpgniZgZl, — RpZp — ZpZi. orRp = ^ gmZgZi, — Zp 1—284 (1) Dividing both sides by Zl, we have mum equal to fiZg = which is ap- proached as Rp approaches zero. At the other ideal extreme, Zp and approach zero and the Rp/Zi, ratio becomes very large. Now, a large Rp/Zt is desired, but some compromise must be made, since the Q of the feed-back circuit becomes increasingly small as Zg approaches Zpg in magnitude. A rigor- ous treatment of the problem to find that relation between Rp/Zl and Qf that provides an optimum frequency stability would require that a complete equation of frequency stability be established and that those impedance relations be determined that produce a minimum frequency deviation for small changes in the circuit parameters. Equation (2) in paragraph 1-288, which is a first order expression for the fractional change in frequency for a change in Rp, indicates that the percentage deviation in- creases directly with the first power of Cp, and inversely with the second power of Rp. This sug- gests that the stability increases as long as Cp/Rp* decreases with an increase in Cp, and is a maxi- mum at the value of Cp, if existent, at which this ratio begins to increase. Such an approach will not be attempted here. Unless all the characteristic curves of a vacuum tube are available, so that either /x, Rp, or gn, can be used as an independent variable to eliminate the other two from the equa- tions, concrete conclusions cannot be reached con- cerning the optimum design of an oscillator using that particular tube. A more qualitative analysis is presented below, and although the indicated optimum relations cannot be considered conclusive, they can serve as first approximations. All imped- ance, current, and voltage symbols given below are considered positive and undirected. Now, Ep = Ip Zl Rp Zp ^ 1 Zl gmZgZL — Zp gmZgZL _ Our present concern is to seek the largest practical value of Rp/Zl, so that the phase of Ip will be least affected by small changes in Rp. Now, Z* «« Zp, — Zp, where Z^ represents the predetermined crystal impedance, which is approximately equal to X,. Also, Zl «= Zp*/Re. On substitution of these values in the equation for Rp/Zl, it is found that Rp R. ^ R; Zl gmZpZg — Re g,„(Zp(Zp — Zp*) — R* 1—284 (2) It can be seen from equation (2) that for oscilla- tions to be maintained gm (ZpgZp — Zp’) must be greater than R,. A maximum Rp/Zl ratio is ' approached as the product gm(Zpg Zp — Zp’) ap- proaches the value of R*. Of course, it is impossible for the denominator in equation (2) to be actually equal to zero, for then Rp would be infinite; but it is plausible to assume that a denominator much smaller than the value of R. can be realized. Thus, we can write (optimum) g„ = R. ZpgZp - Zp* Re ZpZg 1—284 (3) The more nearly this equality is approached, the greater will be the frequency stability. The ques- tion arises, is it preferable to seek this equality with a small or a large value of g„ ? Assuming that WADC TR 56-156 12 « £ (Rp/Zl) >> 1, the equation, Ip = , can be written approximately Ip = f»Eg/Rp = gmE,, or gm = Ip/Eg. A large g„ means a large r-f plate current for a given excitation voltage. This would be desirable from the point of view of maximum output, but an examination of the denominator in equation (3) shows that a large transconductance means that the plate impedance, Zp, or the grid impedance, Zg, must be made small if the equation is to hold. A small Zg (large Z„) means a large Zl and also a large E„/Eg ratio. Both consequences are incompatible with a large Rj/Zl ratio. The former is obviously so, and the latter is implicitly so because the ratio of Ep/E, times ^ k = must equal /i. Every increase in Ep/Eg erefore requires an approximately proportional decrease in the Rp/Zl ratio, insofar as n can be assumed to remain constant. On the other hand, a large Zg and small Z, permits a large Rp/Zl ratio and has the additional advantage of permitting a given excitation voltage with a minimum crystal current. It is under these conditions that equation (3) will be most nearly exact. There are serious disadvantages, however, when operating at a max- imum Rp/Zl ratio; the most important of which is that the Q of the feed-back circuit rapidly de- creases as Zg is increased, since Q( < Zpg — Zg R. Also, the voltage output is weak, and has a tendency to instability. This will be discussed more fully later. Since the excitation voltage is stronger for a given crystal current, the grid losses increase propor- tionately and may no longer be negligible. Fur- thermore, unless the tube is operated class C, the power efficiency is very low. These last mentioned disadvantages, nevertheless, are minor compared with the effect on Qf. The minimum effective Q of the average crystal unit when operating into its rated load capacitance is not unduly large. A grid- to-cathode reactance equal in magnitude to three- fourths Xe reduces Q, to one-fourth Q«. Since the purpose of a large Rp/Zt ratio is to permit the entire plate-circuit impedance (Rp -f- Zl) to ap- pear as nearly resistive as possible, the better stability risk is to operate the parallel-resonant oscillator with small rather than large Ip and gp,. Since we are assuming that >« Ip/Eg, it can be seen that the smaller the value of gm, the smaller is the r-f plate current for a given excitation volt- age, or, for a given plate current, the smaller the value of g«, the greater the excitation voltage. The problem becomes one of determining what capaci- tance ratio, Cg/Cp, permits the smallest possible Section I Crystal Oscidatars gm. By equation (3), gm is a minimum when the denominator of the right-hand term is a maxi- mum. Since the impedance of the crystal unit, Zp„ is to be held constant, (Zp -f- Zg) is also a constant. Thus, the product Z,Zg can easily be shown to be a maximum when Zp = Zg = Zpg/2 1—284 (4) A maximum operating Rp and a minimum Ip with a given excitation voltage can thus be obtained when the capacitance and voltage ratios are Ep/Eg = Cg/Cp = 1 It is quite fortunate that gm has a minimum value. At all other operating values a small change in Q the jr ratio causes gm and Rp, and hence the fre- quency, to change. At the minimum gm the rate of change in the tube parameters is necessarily zero, so that the stability in this respect is a maximum. When the more exact equation 1 — 289 (2) is used instead of equation (1) above, and when /t/Rp is substituted for gm, it can be shown that •D _ Xcp Xe -1- Xcp Xcg 4- /iXcp Xcg ^ r: 1—284 (5) Note that Rp, as long as is constant, is inversely proportional to R,. Now, approximately Xcp = - (X. -h Xcg) Substituting in equation (5), Rp becomes a func- tion of X,, Re, n, and Xc*. Assuming that the first three parameters are constant, it can be shown that Rp is a maximum when X. (m -h 2) 2 (/i -H 1) 1 + P+ 1/ 1—284 (6) If (/* + 1) >> 1, equation (6) states approxi- mately the same conditions as does equation (4) . If /I is small, equation (6) should be accepted as the more accurate in computing the optimum Cg/Cp ratio, since a minimum gm coincides with a maximum Rp if the d-c plate voltage is kept con- stant. The capacitance ratio and values corres- ponding to equation (6) are Cg/Cp ft M 2 1—284 (7) WAOC TR 56-156 129 Section I Crystal Oscillators or C. and Cp 2C, (m + 1) (m + 2) 2C, (m + 1) 1—284 (8) 1—284 (9) Under these conditions the excitation voltage be- comes greater than the voltage across the plate load by a factor of — , and the following addi- tional relations hold: (max) Rp = (m + D Zl 1—284 (10) (min) g„, = ^ 1—284 (11) tVp As a practical consideration in design as well as for the sake of simplicity in discussion it is con- venient to assume that the optimum Cg/Cp ratio is equal to one rather than — However, in inter- /I j preting the equations above, a word of caution is necessary. Returning to equation (2), it will be seen that the maximum to be sought for Rp/Zi, is a “practical,” not a “mathematical” maximum in the sense that a curve of R„/Zi, rises to a peak and then decreases. The curve of equation (2) plotted against Zp passes from positive to negative infinity as the denominator passes through zero and thus is dis- continuous at that point. However, for any given value of gm sufficiently large for Rp/Zt to be posi- tive, the curve does have a true minimum, not a maximum, at the point where ZpZg is a maximum. To avoid confusion as a result of this apparent contradiction, it is important to recall that the “practical” maximum is to be sought by making equation (3) as nearly true as possible, and not by the process of making Zp = Zg. This latter con- sideration is in the interest of over-all stability and maximum activity (if measured by the d-c grid current) for a given d-c plate voltage and load capacitance. Another point that should be well understood is that the minimum g„„ minimum n, and maximum Rp, are all coincidental. From the point of view of frequency stability the real in- terest is in the maximum Rp. From equation (10) it can be seen that the magnitude of the maximum Rp will increase with but remember that this value of /i is the minimum obtainable with a given tube and Ep. As will be discussed more thoroughly in paragraphs 1-294 and 1-295, an oscillator vac- uum tube cannot be operated so that is the maxi- mum possible without the risk of amplitude instability. Thus, class- A operation where the tube is operated only along the straight portion of the E,Ib curve is not feasible in gridleak oscillators. Understand that if equation (7) holds, equations (10) and (11) automatically hold. in each equa- tion is the effective n when equilibrium is reached and is not the starting It is the minimum n that can be obtained as long as the crystal resistance. Re, and the total load capacitance, Cg, remain con- stant. Because Rp, gm, and /i all pass through ex- tremes at the optimum capacitance ratio, it might be thought that the operating conditions are more ideally unique than they actually are, because the instantaneous rate of change for all the tube parameters with the capacitance ratio is zero under these conditions. Remember, however, that these maximum and minimum values apply only in the event that the total C, remains constant. An independent variation in Cg or Cp will cause the frequency to change, and the tube parameters will vary. For instance, gm will tend to vary directly with both Cg or Cp. Only when Cg and Cp are adjusted simultaneously so as always to maintain the same total load capacitance will the instanta- neous changes in the tube parameters be zero as the capacitance ratio is varied through its opti- mum value. 1-285. If equation 1 — 284 (3) is expressed as a function of Zp*, by substituting from equation 1 — 284 (4), the minimum value of g^ becomes, approximately, 4R (min) g™ = ^ 1—285 (1) Since Zpg is the crystal impedance, approximately equal to Xp, the minimum value of gn, can be ex- pressed as (min) g„. = ^ 1—285 (2) where PI is the performance index. Using a one- to-one capacitance ratio and a vacuum tube of high Rp, equilibrium will be reached at the value of gm defined in equation (2). Such operation generally provides the maximum frequency stability in a Pierce oscillator. In estimating the value of PI, Xp is numerically equal to the reactance of the rated load capacitance, and Rp must be assumed to be the maximum permissible effective resistance according to the military specifications of the crystal unit being used. WADC TR 56-156 130 CAPACITANCE RATIO, C,/C„ FOR GREATER OUTPUT IN PIERCE CIRCUIT 1-286. Where a maximum output consistent with the minimum frequency-stability requirements is desired in a Pierce oscillator, the Cg/Cp ratio can be increased and a vacuum tube providing a large transconductance and a large amplification factor should be used. The first consideration is that Ig must not exceed a value that would cause the power dissipation in the crystal unit to exceed the specified drive level. If the output of the oscillator is capacitively coupled to the grid of a buffer ampli- fier, the output power becomes a minor consider- ation compared with the output voltage. If this voltage is to be a maximum for a given tank cur- rent, the plate impedance Z,, must be a maximum. This means that the capacitance ratio, Cg/Cp, must be as large as practicable. The larger this ratio, however, the smaller will be the excitation voltage for a given Ig. The smaller the excitation voltage, the smaller will be the gridleak bias, and conse- quently Rp will be less whereas gn, and will be greater. An examination of equation 1 — 284 (3) reveals that the required magnitude of gm becomes very large as Zp approaches the value of Zpg of the crystal unit. Where a relatively large frequency deviation can be tolerated a large Rp may not be necessary, so that increased voltage outputs can be obtained with tubes of high transconductance at low d-c plate voltages. In any event the Cg/Cp ratio can never exceed the amplification factor of the tube, nor can the r-f plate voltage be greater than, nor equal to, the voltage across the crystal unit. It should be understood that the higher volt- age outputs are only to be had with a large Cg/Cp ratio because of the limitations on the crystal drive level and the load capacitance, Cg, and are not due to the fact that the Pierce or Colpitts type of cir- cuit is inherently more active when the Cg/Cp ratio is a maximum. To the contrary, with a fixed C., maximum amplitude of oscillations is to be obtained when Cp = Cg = 2C,. Where a larger than minimum power rather than voltage output is required, this can probably best be achieved when Cg/Cp lies between 1 and 2, and in this case a larger g„, is necessary to maintain oscillations. When a relatively large power output is required, the Pierce circuit should not be used. HOW TO ESTIMATE THE FREQUENCY VARIATION AND STABILITY OF A PIERCE OSCILLATOR 1-287. In paragraph 1-243 it was shown that the frequency-stability coefficient, Fx., of the crystal unit is defined as the percentage change in X, per Saction I Crystal Oscillators percentage change in frequency. By equation Op t 1 — 243 (1), Fxe = ^ ^ . The reciprocal, 1/Fx„ is thus equal to the percentage change in frequency per percentage change in reactance. Since X, is equal numerically to the reactance of the total load capacitance C„ the fractional change in the load reactance multiplied by 1/Fx,. will give the frac- tional change in frequency. Thus, ^ ^ _ J_ ^ _ J_ f “ w " Fx. ■ Xg " Fxe ■ Cg 1—287 (1) In the Pierce circuit, if it can be assumed that the interelectrode plate-to-grid capacitance and the grid and output losses are negligible, Cg will equal r> } , r where Cp' = Cp Cp. (See figure 1-121 '-'p + t/g (C).) If it is desired to find the fractional change in frequency for a small change in, say Cp', equation ( 1 ) can be used by expressing dCg as a function of dCp', thus : dCg = dCp' • Qv/n With _ Cp'Cg, - “ Cp' + Cg then dCg ^ Cg^ dCp' (Cp' -H Cg)^ and dCg Cg^dCp' / Cp'Cg Cg " (Cp' -t- Cg)^ / Cp' -h Cg ^ Cg dCp' ^ _c^ , Cp' (Cp' + Cg) (Cp')^ On substituting in equation (1) : f 1 Fxe c, (Cp')^ dC„ Likewise, In the event that Cp' =r C, = 2Cg Equations (2) and (3) become df dCp' f “ 4 Fxe 1—287 (2) 1—287 (3) 1—287 (4) WADC TR 56-156 131 Section I Crystal Oscillators ^ = - dCr f " 4Fxe 1—287 (5) 1-288. When a more detailed expression of the fre- quency deviation is desired C,,' can be replaced by (C„ -|- C„), and Cn, in turn, can be expressed as a function of its variables. A rigorous analysis of the effect of each parameter upon the frequency would be quite involved. Probably the simplest ap- proach for determining the frequency deviation due to a change in some particular circuit param- eter would be to begin with an appropriate equation in paragraph 1-287, and express the differential element as a function of the differential of the particular circuit parameter. This is the method that was used when dC, in equation 1 — 287 ( 1 ) was expressed as a function of dC/ and of dC* in equations 1 — 287 (2) and 1 — 287 (3) , respective- ly. As an additional example, suppose4;hat it is de- sired to determine approximately the frequency deviation due to a change in Rp of the vacuum tube. Let it be assumed that C* = 2Ci. The most appro- priate equation to begin with is 1 — 287 (4), since dC„' can be expressed as a function of dRp. The problem is to determine the function that gives the change in C,,' due to an infinitesimally small change in Rp, and to substitute that function for dC„ dCp' in equation 1 — 287 (4) . Since dCp' =:-gj^'dRp, the first step is to determine dCp'/dRp, and then simply to multiply this by dRp. Now, by equation 1—281 (16), Cp' = Cp - Cp/Q,^ Cp Rp Rp + Re so dCp'/dRp = CpRe/(Rp + Re)' or dCp' = — • dRp 1-288 (1) Substituting this function for dCp' in equation 1—287 (4) ; df Cp R, dRp f 4(Rp -H RJ' ■ Fxe 1—288 (2) Note than an increase in R„ causes a decrease in the frequency. It should be well understood that the equations above are only rough approxima- tions. For example, one of the approximations in equation (2) is the assumption that Rp and R, are independent variables, which, of course, is not true. However, the direct effect of a change in Rp upon the frequency can be considered much greater than the indirect effect due to a change in R, re- sulting from the initial change in frequency. The differential element, dRp, in equation (2) can, in turn, be expressed as a function of an infinitesimal change, dEp, in the d-c plate voltage. The general equation for this function is dRp = 4^ • dEp. How- ever, the derivative term, dRp/dEp, will be quite difficult to determine, except by experiment, since it depends upon both the tube and circuit char- acteristics at the operating voltages. For its mathematical expression the principal consider- ations would be the change in Rp with a change in Ep, assuming a constant grid bias, the change in grid bias due to the change in plate voltage, and the change in Rp due to a change in grid bias, as- suming a constant plate voltage. In using the equations above it is only necessary to substitute small finite changes in the independent variable for its differential. For example, if the input ca- pacitance were to decrease an amount aC, = the fractional change in frequency would be given approximately by equation 1 — 287 (3) if we were to substitute — for dCg. ENERGY AND FREQUENCY EQUATIONS OF PIERCE CIRCUIT AS COMPLEX FUNCTIONS OF LINEAR PARAMETERS 1-289. It is beyond the scope of this handbook to present the more rigorous analyses of the various oscillator circuits. These can be obtained from the various reference sources listed in the index. Actually, even when following the more explicit equations, so many approximations must be made for the sake of simplicity, and so many unknowns are involved, such as stray circuit capacitance, that the final solutions can rarely be considered more than general indices of the actual circuit con- ditions. If maximum mathematical exactitude is desired in determining the frequency and activity characteristics of an oscillator, the analysis should be performed by differential equations assuming nonlinear parameters. Such equations are quite involved and are rather difficult to interpret qual- itatively. If a moderately rigorous analysis is de- sired, the equivalent circuit in figure 1-119 (B) may be assumed to have linear parameters, as has been assumed in our previous discussion, but in- stead of handling the impedances as real numbers, to represent them by complex functions. For the Pierce circuit Zp = jXcp Zg = jXcg Zpg= R. + jX, WADC TR 56-156 132 An equation expressingr the conditions for oscilla- tion is derived very similarly to the method fol- lowed in paragraph 1-284, except that I, is expressed by the exact equation Section I Crystal Osdllators necessary for the feed-back energy to be sufficient and stable — in other words, for the loop gain to equal unity. The equation for the imaginary part when made equal to zero can be expressed as: I = ” ‘ Zp. -h Z, rather than by the simplifying equation I. = Ep/Zp For either the Pierce or Miller circuit, the condi- tions for stable oscillation are expressed by the equation Xcp + Xc, + X. -h = 0 1-289 (3) When this equation holds, the loop phase shift is zero. Whereas equation (2) is said to define the feed-back energy requirements, equation (3) is said to define the frequency requirements. Note that in equation (3), the term is equivalent to the reactance of a dynamic capacitance -Rp Zp (Zp, -b Z.) gm Zp Zj -f- Z, 1—289 (1) /-I Rp 1—289 (4) where Z, = Zp -|- Zpg -f Z^ Elquation (1) is similar to equation 1 — 284 (1) except that in the derivation of equation (1) I, is expressed by its exact func- tion and the impedances represent complex quan- tities. When the impedances are expressed as complex functions, the right-hand side of equation (1) can be reduced to the sum of a real quantity and an imaginary quantity, each with the dimen- sions of impedance. Thus, Rp = R 4- jX However, since Rp is not, itself, reactive, the imaginary term, jX, must equal zero, and the real term, R, must equal Rp. In this manner two equa- tions, X = O and R = Rp, involving the same variables are obtained, both of which must hold if stable oscillations are to be maintained. A mini- mum of two equations is necessary, since there are two independent functions to perform. One func- tion is to fix the frequency so that the excitation voltage is properly phased, and the other function is to ensure that the feed-back energy per cycle is exactly equal to its dissipation per cycle. If both sides of equation (1) are divided by Rp, the right- hand side again reduces to the sum of real and imaginary parts. The real part must be equal to 1, and the imaginary part must again be zero. It can be shown that the real part becomes, after multiplying through by Rp : which can be imagined to be in series with Cp. The capacitance of the combination becomes Cp Cd _ Cp Rp Cp + Cd Rp -l* Re 1—289 (5) which is exactly the same as the small-Rp value of Cp' in equation 1 — 281 ( 18) . Ca is thus a positive series dynamic capacitance equivalent to part of the negative parallel dynamic capacitance Cp. Equation (3) indicates that as Rp increases in- definitely Xrp -|- X( g + Xe -♦ 0. This is not to be interpreted as meaning that the tank circuit ap- proaches a parallel-resonant state as a limit or that the total dynamic capacitance approaches zero. Actually, even if the sum of the first three reactances did equal zero, the tank would not be at resonance because of the presence of R, in the feed-back arm, and a dynamic capacitance would need to be effectively present. What equation (3) does show is that, as Rp increases indefinitely, the frequency becomes entirely determined by the tank-circuit parameters. In the limit, Xc* X, = — Xcp. As this state is approached, the impedance of the feed-back arm can be represented as Zpgc R« “1“ jXpgc Re jXcp The impedance of the tank circuit is Xcp (X. 4 - Xc.) Re ~ gm Xcp Xc. 1—289 (2) This equation, rather than equation (1) is the real equivalent of equation 1 — 284 (1) and serves the same purpose in that it defines the conditions Zl = Z R. 7 pgc + Zp - -h jXc p (Re ~ jXcp) jXcp Rp jXcp *4“ JXcp The real component, XcpVRe, is equivalent to Rzp, WADC TR 56-156 133 Section I Crystal Oscillators and the imaginary component, jXcp, is equivalent to Xz,,. Thus, the tank circuit does not appear as a pure resistance in the limit, but approaches an effective Q = |Xcp I R. Rz. XcpVRe I Xcp 1 Re I X, - Xc, I Q, This is the same effective Q. as was determined qualitatively from the viewpoint of phase angles. The term ReXc„/Rp in equation (3) is therefore not to be interpreted rigorously as the total dy- namic reactance, but as that part of the dynamic reactance that is a function of the tube param- eters. At all times, the total l(^d reactance across the crystal terminals is X, = — I/mC, = Xc* + Xcp' which is always slightly greater than (Xoe -j- Xcp). CAPACITIVE ELEMENTS IN DESIGN OF PIERCE OSCILLATOR 1-290. It was declared that equation 1 — 289 (3) defined the frequency requirements of a Pierce oscillator. Since the frequency and the value of X, and Xj are effectively predetermined constants, the primary problem involved in the solution of equation 1 — 289 (3) lies in determining the values of the lumped capacitances that must be inserted in the circuit to provide the correct value of Ci. In the average Pierce circuit the dynamic capa- citance can be considered negligible when com- pared with the total load capacitance, so that Cx can be assumed to equal Cp and Cg in series to a first approximation. With Cp and C* decided upon, approximate values of gm can be had by equating the denominator of equation 1 — 289 (2) to zero and solving for the transconductance. The maxi- mum and minimum equilibrium values of gm coin- cide with the maximum and minimum expected values of Rp, respectively. Next, assuming that gm Ip/Eg, and that E„ = IgXcg, where I* is the crystal current, a vacuum-tube and plate voltage are chosen which will provide a maximum Rp, but which will not cause a crystal unit of any expected Re to be overdriven. To determine the lumped ca- pacitances that must be added to provide the cor- rect values of Cp and Cg, it is first necessary to (A) C9 Figure 7-124. (A) Conventional Pierce oscillator and butler-amplifier circuit. (B) Static capacitances of circuit (A). Cp = interelectrode capacitances. C„ = distributed capacitances of wires WAOC TR 56-156 134 Section I Crystal Oscillators know the values of the stray static capacitances in the circuit. These stray elements effectively create a lower limit to the capacitance across the crystal unit. Obviously, they must not be allowed to exceed the total specified load capacitance. Pref- erably, they should be as small as possible and not be so distributed that an optimum capacitance ratio, Cg/Cp, cannot be achieved. The static capaci- tances to be considered in a conventional Pierce oscillator are illustiated in figure 1-124. Before the optimum values of 0, and Ci can be determined, an experimental circuit should be constructed with the various leads and circuit components reasonable facsimiles of those intended in the final production models. With C, and C3 omitted, the static ca- pacitances between plate and grid, plate and cath- ode, and grid and cathode can be measured. C, and C3 can then be computed to provide the total required load capacitance. 1-291. In the early days of crystal oscillators, and even today where the oscillator is not required' to meet a frequency tolerance less the 0.02 percent, the choice of grid and plate circuit capacitances was largely a matter of trial and error. Usually, (Ai (Bl figun I- 1 25. Variation of activitY and froquoncy in Piorco o$€lilator at plate-circuit capacitance it increated the final choice was based upon the combination that provided the maximum activity for a given d-c plate voltage. For example, if the grid-to- cathode capacitance, C*, of a Pierce oscillator is held constant while the plate load capacitance is varied from a minimum to a maximum, an activity curve, as measured by the gridleak current, is obtained similar to the one illustrated in figure 1-125 (A). Formerly, it was not unusual for the optimum capacitance to be considered a value slightly greater than that at which the activity is a maximum. In military equipment the principal consideration now is to ensure a given total load capacitance. As shown in figure 1-125 (B), in- creasing the plate-circuit capacitance causes the frequency to decrease. As the frequency decreases, so does X, of the crystal unit, and at only one point along the curve will the crystal unit be operating into its rated capacitance. As stated previously, the circuit must provide the specified capacitance if there is to be an assurance that the required fre- quency tolerance is met when one crystal unit is replaced by another of the same type and nominal frequency. In an exceptional case, the most im- portant consideration may be to maintain a fixed frequency relative to some frequency standard. For this purpose small variations in the load ca- pacitance that can be made manually should be possible, but care should be taken that an operator is not to be able to vary the total more than is just sufficient to allow for a frequency variation equal to the bandwidth of the tolerance range. Other- wise, there can be a risk of overdriving a crystal unit, or of continuing in operation a defective crys- tal or other circuit component that should be re- placed before a complete breakdown is threatened. Measurement of Stray Capacitances In Pierce Circuit 1-292. In order to measure the stray static ca- pacitances in a circuit such as that shown in figure 1-124, the lumped grid and plate capacitances, Ci and C3, as well as the crystal, should first be re- moved. The remaining elements can be assumed to form a three-element network as shown in figure 1-126. If an r-f choke is connected in the circuit, the frequency of the Q meter should be approxi- mately the operating frequency of the oscillator. The measurements are made with all vacuum-tube voltages off. The three capacitances in figure 1-126 represent three independent variables, so that a minimum of three measurements is required to determine their values. A fourth measurement is desirable as a check on the accuracy of the first three. Any combination of measurements can be WAOC TR 56-156 135 Section I Crystal Oscillators p o © CpG *Cpg + Cqc *Cqc + C9C Cpc Cgc-fCpc Cpg Cpc Cp9 4 - Cpc © Cpc *Cpc + Cpg Cpc Cp9*fCgc * (4-9) CxY ’C«» + (Cyl OR C«ll Figure 1-126. Equivalent network Formed by the stray static capacitances of a vacuum-tube circuit. A mini- mum of three different measurements is required. The individual capacitances can be determined by solving simultaneously any three of the nine equa- tions above when the respective terminal capacitances are known * Generalized equation for a measured terminal capacitance (Cxr = Cpr„ Cor, or Cpr) when either Cry or Cyz, one of the two respective series-branch capacitances, is shotted out. made, the only restriction is that no two meas- urements are the same and that each of the capacitances is involved in at least one of the measurements. Three different measurements are possible between any one pair of electrodes. Thus, in measuring the capacitance between grid and cathode, the circuit is unchanged for the first meas- urement, the plate can be grounded for the second measurement, and shorted to the grid for the third. For the three measurements : (1st) Cgc = C*e -F (2nd) Cgc = Cgc -|- Cpg (3rd) Cgc = Cgc -f- Cpc Theoretically, these three measurements could be sufficient, but one or more additional measure- ments are needed as a check in the event that the Q-meter leads or the shorting wires have signifi- cantly affected the readings. With Cp,, C,,,, and Cgc determined, the lumped capacitances for both grid and plate circuits can readily be determined. BIAS VOLTAGE OF PIERCE CIRCUIT 1-293. Oscillator bias voltages are not as critical as are those of other types of vacuum-tube circuits, for usually it is neither wave shape, maximum out- put, nor maximum power efficiency that is of most importance, but simply a constant fundamental frequency. For this purpose, the Pierce circuit can be designed to obtain a maximum Rp with a mini- mum drive level, and the optimum bias voltage will be the one that most nearly answers the need. As a general rule, in the case of a vacuum tube that is not being driven to saturation, the effective Rp increases as the bias becomes more negative. If the bias is sufficient for the tube to be operated class B or class C, so that the plate current is cut off for an appreciable part of each cycle, such oper- ation greatly improves the power efficiency. How- ever, a bias developed across a gridieak resistor can never exceed the peak excitation voltage dur- ing stable oscillations. The excitation voltage, in turn, is limited by the current Ig, which must not exceed a value that would cause the losses in the crystal to exceed the permissible limit. When i , is maximum, the limiting crest value of Ig is Igm , where P™ is the maximum permis- sible drive level in watts, and R^m is the specified maximum effective resistance when the crystal unit is operating into its rated load capacitance. The maximum grid bias is thus (max) Ey = - 1-293 (1) \ ^ein where Zg ^ is the r-f impedance between grid and cathode. It was found earlier (paragraph 1-284, equation 1 — 284 (8) ) that a maximum Rp is to be had when Cg = — ''.IC . or approximately when Cg = Cp 2Ci. Equation (1) under these conditions becomes (max) Ee = - , 1-293 (2) 2uC,VRy,„ Equation (2) is quite important in that it shows that the maximum grid bias obtainable with a maximum R^ is fixed by the crystal specifications. The maximum E.. for any given crystal unit may be obtained from equation (2) by substituting the actual Re for R,.„,. The proper anode voltages for a given vacuum tube, or vice versa, to provide a re- quired output are consequently also effectively predetermined by the crystal specifications. For maximum stability with any given vacuum tube in WADC TR 56-156 136 Section I Crystal Osciiialors a conventional Pierce circuit, C( and Cp should each be made equal to 2Ci, or if the amplification factor is small, C,/Cp should be made equal to - ^ with the total capacitance in series made to equal C^. With the capacitance so determined, the plate and screen voltages can be adjusted to give the desired output and excitation voltages. Since Ep Ep, the output voltage is also effectively lim- ited by the crystal specifications. With Cg = Cp = 2Ci, (max) Ep = .707 (max) |E,.|, where (max) Ep is the maximum r-ih-s value of Ep, and (max) E^ is given by equation (2). FIXED BIAS FOR PIERCE OSCILLATOR 1-294. It is not conventional to employ a fixed bias in a crystal oscillator, although it can be done — even to advantage in some cases. An r-f choke can be substituted for the gridleak resistance, thereby reducing the grid losses to an absolute minimum as long as the excitation is insufficient to overcome the bias and cause grid current to flow. In order for stable oscillations to be maintained, an increase in excitation roust cause a decrease in amplifica- tion, and a decrease in excitation must cause an increase in amplification. When using a fixed bias, the choice of operating voltages is much more re- stricted than when employing gridleak limiting. Because of the more critical operating conditions, the replacement of one crystal unit with another having a different resistance may require addi- tional circuit adjustments. If a fixed, class-B or class-C bias is used, a slight decrease in the ampli- tude of oscillations normally leads to the oscilla- tions dying out all together. This is because the average amplification of the positive alternation of each cycle increases and decreases directly with the amplitude instead of inversely. For instance, with a class-C fixed bias, a decrease in the ampli- tude of one cycle would mean that the tube is cut off during a larger fraction of the succeeding cycle, thereby further decreasing the average amplifica- tion. On the other hand, if oscillations were once started, the tendency would be for the amplitude to build up until limited by grid and plate satura- tion. Only if limiting is provided by nonlinear ele- ments, such as thermistors or varistors, in the external circuits is class-B or class-C fixed-bias operation possible if the tube itself is not to provide the limiting action. A fixed bias can be used if the tube is driven to saturation each cycle, but such operation is not practicable unless the utmost power is required from the oscillator, and in any event should not be attempted with the Pierce circuit. The only feasible application of a fixed bias ib Figure 1-127. Point on It curve at which stable oscillations can be achieved with fixed bias. For minimum grid losses, the peak of the maximum excitation voltage must not exceed the fixed in the Pierce oscillator is to operate the tube just above the knee of the E,.Ib curve (see figure 1-127) . (E,. and Ib represent no-signal, d-c values of grid voltage and plate current respectively.) At the indicated operating point a slight decrease in the activity results in the average amplification of the negative alternation of the excitation cycle being greater, whereas the amplification of the positive alternation remains essentially constant. Thus, the over-all amplification of the weaker cycle is greater than that of the stronger, and a stable equilibrium is possible. An r-f choke would normally replace the gridleak resistor. Such operation practically eliminates the grid losses as long as the peak exci- tation voltage does not exceed the bias. Theo- retically, then, the fixed bias permits a maximum Q in the feed-back circuit, and in this respect aids the frequency stability. There are, however, prac- tical difficulties involved. If the C«/C,, ratio of the external circuit is such that the oscillator is to operate at the minimum g„„ it may be difficult to find a vacuum tube that provides the desired trans- conductance when operated at suitable voltages just above the knee of the EJb curve. The low transconductance can readily be achieved by using a remote-cutoff tube, but the amplitude stability will be more critical since the amplification of the positive alternations increases with amplitude, thereby tending to annul the limiting action of the negative alternations. Variations in the circuit capacitances have the following effects, which are essentially the same as those that occur with grid- leak bias except that the amplitude variations are more pronounced in the fixed-bias circuit. A small decrease in the capacitance ratio, say by an increase in C,„ would mean that the voltage ratio, Ep/E* = C,/Cp = = g,.. Z,, WADC TR 56-156 137 Section I Crystal Oscillators must become smaller. Since Ep/'Eg varies inversely with Cp whereas Z,, varies inversely with C,,', an increase in C,,, as long as Rp>> Z),, requires gm to increase proportionately in order that equilibrium may be re-established. In other words, an increase in C|, must cause the excitation voltage to decrease. Conversely, a decrease in C,, causes g„, to decrease and the excitation voltage to increase. Rp varies in a direction opposite to that of g„„ but the per- centage change in Rp is not as large as the per- centage change in g,„. The change in g,„ is approxi- mately proportional to the change in Cp. If Cp is constant, but C,. is varied slightly, Zi. will be ap- proximately constant, so that g„, must also vary directly with Cp. in order for equilibrium to be maintained. Therefore, the changes in the excita- tion voltage with changes in C^ are similar to those with changes in C,,. When Cp and Cg are varied independently the total load capacitance Cj changes, and hence the frequency and X, also change. If, during tuning adjustments, Cp and Cg are varied so that the same total capacitance is always maintained, then, in the region where Cp ^ Cg the change in Cp is approximately equal but opposite to the change in Cg. In this case g„„ which has reached its minimum value, tends to remain constant, as do also the excitation voltage and the plate resistance of the tube. It should be under- stood that this optimum condition holds only for variations in capacitance that leave the total load capacitance unchanged. With Cg fixed, g„, must in- crease as the capacitance ratio is varied to either s.de of its optimum value. When Cg/Cp is made greater than one the change in g„, and the excita- tion is greater than when Cg/Cp is made less than one by an equivalent proportion, (e.g., Agn, is greater when Cg/Cp is changed from 1 to 2 than when it is changed from 1 to 1 / 2 -) An increase in g,„ can occur only by virtue of a decrease in excita- tion voltage. Thus, if the grid impedance, Zg, is made larger than the plate impedance, Zp, the crys- tal current, Ig, must necessarily become smaller. Oscillations can thus be maintained with a smaller drive level. Nevertheless, optimum stability gen- erally requires a maximum Rp, which, in turn, coin- cides with a maximum excitation voltage and minimum g,„. 1-295. According to equation 1-284(10), the opti- mum capacitance ratio will automatically cause the oscillator to seek an equilibrium when Rp = PI (/t -f- 1) Zi„ or approximately when Rp = where PI is the performance index. Herein lies the principal limitation of fixed-bias operation of a Pierce circuit. The maximum to minimum values of the PI of a given type of crystal unit can be as much as 9 to 1 for the same frequency — and greater still if the oscillator is to operate over a wide frequency range. It may be difficult to find a tube to provide equivalent variations in the effec- tive Rp unless the excitation voltage is to be so large that it drives the grid positive on the positive excitation peaks. In this case, the principal advan- tage of the fixed-bias— to maintain a maximum Qf and to minimize the variations in the input imped- ance — is lost. Since an increase in Rp must be accompanied by an increase in the r-f plate current operating into a proportionally increased load im- pedance, the tendency will be for the crystal power (approximately equal to Ip* Z|,) to vary directly with (PI)* or^^^ . Unless there is some guaran- tee that the maximum Rg is not to be greater than twice the minimum expected value, some additional form of limiting must be used in the fixed-bias circuit, such as connecting a varistor across the r-f load, to ensure that the low-resistance crystal units are not over-driven. The fixed bias should not be less than three times that given by equation 1 — 293(2). The vacuum tube (preferably a pentode, because of its low plate-to-grid capacitance and high Rp) should be chosen and the anode voltage determined that permits operation at the lower end of the straight portion of the E,.Ib curve. The opti- mum bias and plate voltages are best established by experiment. The principal problem is to ensure a sufficient output for crystal units of maximum Rg, without overdriving those crystals of minimum Rg. The average crystal unit has an Rg approxi- mately one-third the maximum. An occasional crys- tal unit may have a value of Rg perhaps as small as one-tenth the maximum. It can be seen that a serious obstacle to the use of a fixed bias is that manual adjustments of the operating voltages are necessary when replacing crystal units, unless the plate circuit is to be rather heavily loaded. The output will tend to vary by a large factor from one crystal unit to the next. The same problem is en- countered with the use of a gridleak bias, but volt- age adjustments are not absolutely necessary, even under no-load conditions. A familiarity with fixed- bias operation is helpful, however, in that it aids the understanding of gridleak operation. GRIDLEAK BIAS FOR PIERCE OSCILLATOR 1-296. The importance of having gridleak instead of fixed bias is two-fold: First, it permits a large initial surge of plate current, so that oscillations will build up quickly. If the tube were being oper- ated class B or class C with a fixed bias, the bias WADC TR 56-156 138 would have to be removed before oscillations could build up at all. Second, it ensures a maximum sta- bility in the output. If for any j-eason the excita- tion should increase or decrease, the d-c grid current and hence the bias follows the change, always acting in a direction that tends to annul the original change. When the oscillator is first turned on, the starting bias is zero regardless of the value of the gridleak resistance. Thus, insofar as the initial surge of current is concerned, the value of the gridleak resistance, R„, is not a first- order factor. However, the value of is significant in its effect upon the total build-up time. This effect is considered in paragraphs 1-304 and 1-305, where the conditions most favorable for oscillator keying are discussed. The present discussion considers only the effects of R^ upon the oscillator stability after the oscillations have reached a maximum am- plitude. First, it is desirable that the grid losses be as small as possible, and that they at least can be considered negligible by comparison with the losses in the crystal unit. For a continuous flow of d-c grid current to be maintained, the grid must be positive with respect to the cathode at the positive peak of each excitation cycle. The amount of grid power that is dissipated, the extent to which the grid becomes positive, and the length of the period during which the grid is positive and electrons are flowing from cathode-to-grid depend upon how great a percentage of the total charge escapes through Re during the remainder of the cycle. This, in turn, depends upon the ratio of the period of one cycle to the RC time constant of the grid circuit. This ratio, l/R„Cgf, is seen to be equal to ^”’ Kg The smaller this ratio can be made, the smaller will be the percentage leakage of charge during one cycle, and the more nearly will the bias remain con- stant and equal to the peak excitation voltage. At high frequencies, ratios on the order of 1/50 and smaller are quite easily obtained. With the period of one cycle so short compared with the time it would take 63 per cent of the accumulated charge to leak off, it can be assumed that the bias voltage equals the peak excitation voltage in magnitude. Should the excitation voltage increase, the bias also increases. The peak excitation voltage is E,„, = Ie„, Ze = 1.414 Ie Ze = I Ee 1 1—296 (1) where Eo is the grid bias when ReCj > > 1/f, and Ig,„ is the peak r-f grid-circuit current. The bias voltage is also given by the equation Ee = Rg 1—296 (2) Section I Crystal Oscillators where Ic is the d-c grid current. If an r-f choke having an impedance that is large compared with Rg is connected in series with R^, the r-f voltage across Rg becomes small compared with the d-c voltage. With this arrangement it is readily seen that the approximate grid power expenditure is Pg = Ic E„ = EcVR* 1-296 (3) If the r-f choke is not present, so that the voltage across Rg varies sinusoidally from a peak of — 2Eg„ to a peak of 0 on the positive alternation, the aver- age squared voltage across Rg, equal to ^ f (1 -f sin «t)'‘ d(a)t), is found to be 1.5 Ec*. Thus, in the absence of an r-f choke, Pg = 1.5 EcVR* 1—296 (4) Clearly, if the grid losses are to be held to a mini- mum, Rg must be as large as possible. If an exami- nation is made of several representative crystal oscillator circuits in actual production, it will be discovered that very few employ gridleak resist- ances higher than 100 kilohms, and only an occa- sional value of Rg is found higher than 0.5 meg- ohms. The answer is principally to be found in the fact that the oscillator design is usually a com- promise among several factors: (a) frequency stability, (b) output-voltage stability, (c) output control, (d) operating efficiency, (e) maximum economy in production costs, (f) minimum over-all weight and space requirements, (g) whether or not oscillator is to be keyed, (h) frequency range (i), value of Cg, (j ) whether or not circuit is to pennit switching from crystal to tuned circuit, and (k) the suppression of parasitic frequencies. Either a high or a low value of Rg can improve the per- formance in respect to any one of the factors above, depending upon what is required concerning the other factors. For example, a very large Rg can improve the frequency stability by reducing the grid losses, when only a small output is required. On the other hand, the. same value of Rg could lead to both frequency and output voltage instability if maximum output or maximum operating effi- ciency were required. The effects of Rg relative to various factors listed above are discussed briefly in the following paragraphs. Gridleak Resistance and Frequency Stability of Pierce Oscillator 1-297. The grid Rg can lead to frequency instability in two ways. As Rg is decreased, the grid losses load WADC TR 56-156 139 Section I Crystal Oscillators the feed-back circuit and require that the tank be operated farther away from resonance. Rp of the vacuum tube is reduced and its effect upon the frequency becomes more pronounced. On the other hand, if is increased indefinitely, the oscillator can become self-modulated. This latter effect is more properly classed as a problem of output volt- age stability, and is discussed in the paragraph 1-299. The principal frequency-stability importance of Rg is the degree by which it reduces the effective Qf of the feed-back arm. (Qf is the feed-back qual- ity factor only from the point of view of phase rota- tion.) As defined by equation 1 — 272 (1) current pulse when the grid is positive with respect to the cathode will be considered negligible com- pared with the power losses in Rg per cycle. Assum- ing that no r-f choke is used and that Rg is sufficiently large so that equation 1 — 296 (4) is approximately correct, equation (1) on substitu- tion and rearrangement becomes R*' = 1-297 (2) Since it has been assumed that Rg is large com- pared with Xcg, then Xcg' <=■ Xo* and Eg* = 2(IgXcg)’. On substitution in equation (2), Q _ Qpirc Qk ' Qptc + Q* where Q,,gg is the actual over-all effective Q of the feed-back arm, and Qg is the effective Q of the in- put circuit. In order to analyze the effect of Rg in terms of the equivalent Pierce r-f circuit, we repre- sent Zg by an equivalent reactance, Xcg' in series with an equivalent resistance, Rg', as shown in fig- ure 1-128. The problem now is to determine Rg', such that Ig^ Rg' = Pg 1—297 (1) where Pg is the power expended in the gridleak resistance. The power expended during the short CD ^pgc — © Og = Figure 1-128. X a X C q R« + R’g © Q» = R'a Qpqc Qg OpgC + 0 g fquiVa/enf Pierce feecf-bacfc circuit WADC TR 56-156 3Xo,“ 1—297 (3) If an r-f choke were used in series with Rg, the approximate value of Rg' would be c» R. The values of Qg become (without choke) Qg = I Xcg' I R*' 1—297 (4) Rg 3 Xcg 1—297 (6) (with choke) Qg = Rg 2X, Cg 1—297 (6) The equations for Qpgg are / -il. i I. 1 \ /-V (Xe + Xcg) Rg (without choke) Qpgg = p p , o v 2 Kg K, -|- o Acg 1—297 (7) . -iL 1. 1 ^ (Xe + Xcg )Rg (With choke) Qpgc = p p 1 9 y * XVg IVe “T ^ 1—297 (8) Assuming that Qpgg Qg > > 1, Q( is given by the ap- proximate equations (without choke) Qi = - (with choke) Re 3 X. Xcg Rg Qf 1—297 (9) (Xe + Xcg) Re - 2 Xe Xc, Rg 1—297 (10) 140 In the previous discussions it has been supposed that the grid losses are kept negligible compared with the losses in the crystal, so that a Qr equal to X, + Xo K. * is approximately correct. Where this assumption cannot be made, those equations that define the frequency of the oscillator, such as equa- tion 1 — 289 (3) , can still be used for approximately correct answers if R, is replaced by the appropriate denominator in equation (9) or (10). Thus, the effective frequency-determining resistance of the feed-back arm, can be defined as R,. = R.-l-|3X.Xc,/R,| or R, -h 1 2 X, Xc,/R. | 1—297 (11) On the other hand, in those equations that concern the equilibrium between the energy input and out- put of the feed-back arm, such as equation 1 — 289 (2), the effective feedback-circuit resistance to substitute in the place of R, is the value R( = Re -|- Rg' 1—297 (12) On multiplying equation (11) by Rg/R*. it is ap- parent that if Rg is to have a negligible effect upon the frequency, it must be much greater than ^ • Similarly, if Rg is to have a negligible xV* I effect on the total feed-back power requirements, according to equations (3) and (12), it must be much greater than 3Xog*/Re. If the oscillator is to be operated in the region of maximum R,, of the tube, |Xcg| will approximately equal X,/2, and 3X,X, R. ££. will approximately equal g PI. Under these conditions, a good rule of thumb, if the other operating requirements permit, is to employ a grid resistance equal to 15 times the minimum permis- sible PI ' o T c * R ')* greater, where Re„, = (max) Rg. Grid~Resistance Effects and “Class-D” Operation of Pierce Circuit 1-298. Typical curves showing the effect of the grid resistance upon the frequency and the fre- quency stability of a Pierce oscillator are shown in figure 1-129. The curves in figure 1-129 (A) were obtained for plate voltages (Eb) of 50, 60, and 70 volts under no-load conditions ; the curves in figure 1-129 (B) were obtained for plate voltages of 75, 100, and 125 volts when a load resistance of 5000 ohms was connected across the plate circuit. The curves were plotted from measurements of an ex- perimental Pierce oscillator during a USAF re- section I Crystal Oscillators search project under the direction of E. Roberts at the Armour Research Foundation, Illinois Institute of Technology. The frequency deviation due to a change in R„ with a change in plate voltage is in- dicated by the frequency difference between the points on the curves that correspond to the same values of grid resistance. The four sets of curves for each of the two load conditions were obtained by maintaining the grid-to-cathode capacitance constant and varying the plate-to-cathode capaci- tance. The bottom set of curves in each graph represents the closest approach of the four sets to a Cg/Cp ratio of 1, and the top set represents ap- proximately the largest Cg/Cp ratio (maximum Ep) at which oscillations can be maintained. As is to be expected the frequency deviation due to changes in voltage is greater for the larger than for the smaller values of C,/C„. However, part of this greater deviation is due to the fact that in the cir- cuit from which these curves were plotted, the plate capacitance was obtained from a capacitor paralleled by an inductor. At the minimum effective value of Cp, the capacitor and inductor approach a state of parallel resonance, so that the rate of change in the equivalent plate reactance with a change in Rp is larger than would be the case if no inductor were present. Also, the fact that the total load capacitance facing the crystal is smaller at the higher frequencies contributes to the fre- quency deviation. Partially counteracting this lat- ter condition is the increase in the effective Qp of the crystal and in the Qr of the feed-back circuit as a whole because of the rise in frequency. There are three variables influencing the frequency that are affected by a change in the grid resistance. The first of these is the Q* of the grid-to-cathode im- pedance. As Rg becomes relatively small, Q, be- comes the dominant factor determining the phase of Ep with respect to Eg. As Qg decreases, the tank circuit must appear more capacitive. Thus, the re- actance of the crystal unit in the feed-back arm, and hence the frequency, must increase. This is indicated by the sharply rising tails of the curves in figure 1-129. The second frequency-determining factor affected by Rg is the effective grid-to-cathode capacitance, Cg. As Rg' decreases, Cj effectively in- creases. That is, if Rg and Cg are represented by an equivalent Rg' and Cg' in series, it can be shown that Cg' increases with a decrease in R*. As a result of this effect, as Rg becomes small, the frequency tends to decrease; but, as indicated in the upper sets of frequency stability curves of figure 1-129, when Cp is small compared with Cg, changes in Cg have little effect since the total load capacitance is approximately equal to the smaller capacitance. On WADC TR 56-156 141 Section I Crystal Oscillators the other hand, when the Cg/C,. ratio is small, as indicated by the lower-frequency curves in the fig- ure, the changes in the effective Cg have a measur- able effect upon the frequency. It can be seen at the low values of grid resistance that the effective increase in capacitance greatly diminishes the rise in frequency that would otherwise occur because of the decrease of the grid-to-cathode, Qg. Indeed, the bottom curves in figure 1-129 (A) show that for the particular circuit and crystal unit the two op- posing frequency effects of R, apparently cancel each other when Rg is in the neighborhood of 100,- 000 ohms. This same set of curves indicates that a minimum frequency deviation with plate voltage occurs when the grid resistance is approximately 200,000 ohms. It is with some diffidence that we attempt to explain the reason why this particular value of Rg should provide a point of maximum frequency stability. Rather than interpret the effect as due to a possible optimum ratio, or as due to a possible variation of Rg with plate voltage that tends to cancel the effect of the variation in Rp, it seems more likely that the optimum results are due to a coincidence between the third frequency factor mentioned above and the characteristics of the tube, a 6C4 (triode), that was used in the test circuit. This third frequency factor is the average grid bias, which tends to increase and decrease with Rg, although the variations are not pro- nounced when Rg is large. As the bias increases, so also does Rp, which in turn causes the frequency to decrease. This can best be seen from an examina- tion of the curves at the top of figure 1-129, where the effects of Rg on Cg' are negligible as they affect Figure 1-129. Frequency of tuned Pierce oscillator versus plate voltage and grid resistance for various plate- tuned load capacitances. (Max) represents plate tuning adjustment that provided maximum r-f plate voltage when grid resistance and d-c plate voltage were values indicated by zero reference point. A 7-mc CR-I8/U crystal unit was used, having a PI of 49,000 ohms when operating into a rated load capacitance of 32 /ifJ WADC TR 56-T56 142 Section I Crystal Oscillators the frequency. As R* increases, the bias approaches as a limit the magnitude of the peak excitation voltage. Thus, R„ also rises to some limiting value, causing the frequency to level off to some minimum value. If the plate voltage is increased when the tube is biased below the straight portion of the Eelb curve, one result is a decrease in R,„ which, in the curves of figure 1-129, clearly causes the fre- quency to increase. However, an increase in the plate voltage also causes an increase in the excita- tion voltage, and hence in the grid bias. Thus, it may well be that the point of maximum frequency stability as indicated in the bottom curves of figure 1-129 (A) is the result of an increase in bias with the increase in plate voltage just sufficient to hold R,. constant. A class of operation such that the percentage change in R„ due to a change in Eb is annulled by a percentage change in Rp due to a change in E,., or vice versa, suggests interesting possibilities in stabilizing the plate resistance by methods other than plate-supply regulation. There is no evidence that this type of operation has been investigated, but on the strength that possibilities exist for practical application in oscillator circuits not employing a fixed bias, the name "class-D” operation is proposed. Three subclasses are pos- sible : “D„” where changes in Ei, can occur more or less independently and E,. is the dependent vari- able; “Dj,” where E.. can be considered independent and Eb dependent; and “D, " where E,, and E,, are mutually dependent. Mathematically, this class of operation can be defined by the following equa- tions : (Class D) ARp 1 /0R,.\ Rp Rp \ 9 Eb / AEb £ = const. c ^ 1 b = 0 const. 4E. - AE. 1—298 (1) (Class D.) (AE^ independent) D ae.-|5-ae. 1—298(2) (Class Dj) (AE^. independent) 1—298 (3) (Class D,. s) Equations (2) and (3) both apply. (B) (= 7MC WADC TR 56-156 143 Section I Crystal Oscillators Equations (1), (2), and (3) sufficiently define “class-D” operation, but the parameter, R,„ repre- sents a with-signal average plate resistance, and not the instantaneous or static resistances repre- sented by conventional R„ curves; although the static values would apply if the signal amplitude were small relative to the bias, as would be the case if age were being used. An increase in plate voltage can generally be expected to cause an increase in excitation and grid bias in the conventional oscil- lator circuits. This action, in turn, can be expected to cause an increase in the average R,,. Thus, if “class D” is to be in effect, it is necessary that an original increase in cause R,, to decrease by the same amount as the change in E,. will cause it to increase. For this to occur, the plate characteristic curves must show positive slopes that increase with plate voltage; that is, the Ei>Ib curves must be curving upward in the direction of increasing E,, at the operating voltage, as is quite characteristic of triode curves. Pentodes do not show this char- acteristic if the screen voltage is held constant, since the plate resistance tends to increase with increasing plate voltage. However, if the screen voltage varies with, and in the same direction as, the plate voltage, as can be the case when the two voltages are obtained from the same source, plate characteristics can be achieved similar to those of triodes, but with the advantages of larger values of R,. and an independent variable (E. j) by which the rate of change of R,, with Eb, ( 9R,./9Eb) , can be adjusted. Now there is an additional implied con- dition that must be met if class-D effects are to be achieved in conventional oscillator circuits. This is the requirement that g„, also remain constant. For example, by equation 1 — 289 (2) which is re- peated here o _ Xco (X„ -|- Xcr) ” Re - g,„Xcp Xc* it can be seen that feed-back equilibrium in a Pierce circuit requires that as long as R„ and the external circuit parameters remain constant, so also must g„,. An analogy here is to be found in class-A operation, which is defined by the operation of the tube along the straight portion of the E,.I|, curve, i.e., in a region of constant g„,. Since the principal purpose of class-A amplification is a dis- tortionless output, by implication a necessary re- quirement is that the operation also be in a region of constant R,,. In “class D,” on the other hand, a constant effective R,, is the sufficient definitive con- dition, but in application a constant effective gm is a necessary implication. If g„, is to remain con- stant, equations for Ag,„ similar to (1), (2), and (3) for aRp must hold simultaneously. This can be achieved in one of two ways, or a combina- tion thereof. Assume that the plate voltage in- creases. If Rg is sufficiently small for the positive excitation peak to drive the grid reasonably far above the zero point, an increase in excitation, al- though decreasing the average g„ on the negative alternation, can annul this effect by increasing the average g^ in the region above the zero grid volt- age. For this to occur, the tube would have to be operated at plate voltages low enough for the zero grid point to lie well within the bend of the E,.lb curve. This, indeed, was the state of the 6C4 tube when the curves of figure 1-129 (A) were deter- mined. Presumably, under no-load conditions and a small Cg/C„ ratio, as the grid resistance was decreased, the changes in the positive excitation peaks with changes in plate voltage were just sufficient to maintain g„, approximately constant for values of Rg between lOOK and 200K. If Rg is very large, the effect above is negligible, since any increase in the positive excitation above the zero grid voltage point becomes minor compared with the total increase of the negative alternation. For g,„ to be stabilized when Rg is large, the change in plate voltage must make a change in the cutoff voltage comparable to the change occurring in the excitation voltage. Plate characteristics most prob- ably favorable to “class-D” operation appear to be had with low plate voltages. Unlike the R,, and gm in the conventional classes of amplifier operation, where R„ and g„, can be varied independently, this condition cannot exist in class-D operation of con- ventional oscillators, since R„ and gm are tied to- gether by the feed-back energy requirements at equilibrium. Any condition that would stabilize the one, would automatically stabilize the other. It is only R,„ however, that directly affects the phase of the feedback. Once R,, becomes large relative to the impedance across the tube, the percentage variations in g,,, become very small, so that gm can be assumed to be a constant for all practical pur- poses. It is R|, that requires critical attention if it is to be held constant. As discussed in paragraph 1-342, the curves shown in figure 1-146 strongly suggest the possibilities of “class-D” operation in the case of a Miller circuit. The solution of the “class-D" equations for a given vacuum tube and circuit can probably be approximated graphically, using families of R„ curves versus E^ and £„ or curves of the deviations of the R„Eb and RpE^ curves. If rates of change of E,. with Ep can be ob- tained that provide a solution for equation (1) when the values of Ec and Ep are practicable, the WADC TR 56-156 144 possibility exists that an oscillator of any type, parallel- or series-resonant, using a gridleak bias (or age) can be designed so that for all practical purposes it is independent of small fluctuations in the plate-supply voltage. Empirically, “class-D” operation is indicated at the point or points where the frequency deviation curves of an oscillator change in sign, or where frequency curves, such as those in figure 1-129, cross or touch each other. A full analysis of this class of vacuum-tube opera- tion is beyond the scope of this handbook. It is suggested here only as a possible line of inquiry. Gridleak Resistance and. Output Voltage Stability of Pierce Circuit 1-299. The stability of the output voltage depends largely upon how readily the gridleak bias can follow small fluctuations in the excitation voltage. Imagine, for example, that the vacuum tube is being operated class C, and that after equilibrium is reached the positive peak of a certain excitation cycle happens to be slightly higher than the aver- age. The average bias during the succeeding cycle will thus be slightly more negative than is normal, so that during this period the tube is conducting a smaller fraction of the time, and the peak excita- tion voltage will drive the grid less positive than before. This means that the amplification during this cycle will be less than the amplification dur- ing the preceding cycle. If the Rt.C„ time constant is extremely large compared with the period of a cycle, the bias remains relatively fixed for the dur- ation of several cycles. In which case the peak of several succeeding cycles must rise to progressively lower points on the E,.Ib curve. The oscillations will continue to decay until a sufficient amount of the bias charge of has leaked through Rg to permit oscillations to again build-up. For this reason Rg cannot be increased indefinitely without the risk of the oscillator becoming self-amplitude-modu- lated. As Rg is gradually increased, the amplitude sooner or later begins rising and falling at a radio- frequency rate. If Rg is further increased, the modulation of the output can fall within the audio range. Finally, with extremely large values of Rg, the circuit behaves as a damped-wave blocking oscillator. Now, assume that Rg is infinite. As oscil- lations build up, the bias for each cycle is essen- tially the same as the peak excitation voltage of the preceding cycle. Eventually a peak excitation voltage is attained which causes the bias for the next cycle to be too great for the circuit to be resupplied with all the energy that will be lost during the period of the cycle. If the peak bias is equal to or greater than the cutoff bias, the oscil- Section I Crystal O^illalors lations will die out completely, since any decrease in excitation with class-C bias means a decrease in average amplification. To avoid this possibility, it is important that sufficient electrons escape from the grid so that, at the beginning of the cycle im- mediately following the first peak cycle, the bias will have returned to approximately the same starting point. Expressed in another way, to avoid the intermittent activity, there must be an assur- ance that the positive peak of every cycle will drive the grid positive. This assurance is to be had for all operating conditions if the bias voltage de- creases at a greater rate than would the positive excitation voltage peaks if the tube were cut off for a complete cycle. In practice, the vacuum tube can be conducting in polar opposition to E„ and hence effectively supplying energy to the circuit during the entire negative alternation of an E,, cycle. Nevertheless, if there is an assurance that the bias voltage drops as fast as the peak excita- tion voltage when no energy is being supplied to the circuit, the bias reduction is certainly sufficient if the net rate of energy-loss is reduced by virtue of a variable release of energy by the tube through- out a large part of each cycle. The problem, then, becomes one of first determining the percentage change in the peak excitation voltage that would occur during the period of one cycle if the tube were suddenly cut off. 1-300. At the instant that Ig is a maximum, the voltages across the reactances in the tank circuit are zero, and none of the circuit energy is stored in the capacitances. All the stored energy at that instant is in mechanical form, and is equal to the kinetic energy of the crystal as it swings through its position of zero potential energy. As discussed in paragraph 1-249, this stored energy is equal to I,’L, where R is the series-arm current, and L is the equivalent series-arm inductance. Now, when the crystal appears as an inductance, R is approxi- mately equal to R plus the current, Ro, which flows through the shunt capacitance, C„, of the crystal unit. (Only the unsigned magnitudes of R and Ic» are considered here.) We can say, approximately, that the stored energy is equal to (R -|- Rol’E- As is also discussed in paragraph 1-249, the ratio of stored energy to the energy dissipated per ra- dian, is equal to the Q of the circuit, which in this case is effectively ; b » > or the (Ke -|- Kg 4" Mg eouivalent value — qu aent aiue, (R/-hR,')V’ where RR is the equivalent load resistance when represented as in series with the plate-circuit ca- pacitance, and R is the series-arm resistance of WAOC TR 56-156 145 section I Crystal Oscillators the crystal. If R^' and Rr/ can be considered neg- ligible. the circuit Q will be the actual crystal plate current is cut off for an entire cycle, which is a period of 2ir radians. The fraction of the energy dissipated during this time is approximately equal to 2ir/Q = R/fL, if Q is of sufficient magnitude that the per- centage decrease in current is not large. Since the energy is proportional to the square of the voltage, the equivalent decay in peak excitation voltage is E„„\/R/fL |E..|\/R/^- If E,., the grid bias, whose magnitude only we shall consider, is to de- crease at the same rate, the bias charge, equal to C^Ee, must leak through Rj at an average rate of CgE.VR/fL during the period of one cycle. Thus (min) I, = - p = fC*E, VRTfL (max) Rg 1—300 (1) The maximum safe value of Rg for all operating conditions, if Rg' and Ri/ are negligible compared with R,., according to equation (1) is (max) Rg = VL /Cg VfR 1—300 (2) Since VE' = l/w\/C, where C is the equivalent Q series-arm capacitance, and since C ■=« in the case of partially plated elements, where r is ap- proximately equal to the theoretical capacitance ratio, r,., given in figure 1-95, then, on substitution in equation (2) (max) Rg ‘•'CgV fRc„ 1—300 (3) In paragraph 1-297 it was shown that when Cg = 2Cx, if Rg is to be considered negligible it should be at least 15 times the minimum permis- sible PI. Assume that Rg/ (min) PI = k, (k is not to be interpreted here as a symbol for any quantity other than the ratio defined) and that it is de- sired that k = k„„ its maximum value consistent with equation (3), above. Let it also be assumed that R is approximately equal to R,n., the maximum permissible value of R,, that Cg = 2Cx, and that C„ = C„„„ the maximum permissible shunt capaci- tance specified for the crystal unit. Then, (max) Rg = k.„ (min) PI = Re,„ also (max) Rg = >/2irr / 2u C^V R,,., Thus, (max) Rg/(min) PI = k^ = toC, •%/ 2^ r R em 2^/ &) Com 1—300 (4) or km“ = ■»■«• 1 (min) Xc„ | /2 (min) PI 1—300 (5) It will be found in practice that the limiting values of km given by equations (4) and (5) are normally smaller than the minimum desired value of 15. If this should be the case in an actual circuit, the assumption that the power losses in the grid cir- cuit are negligible can no longer be made, and the actual value of k^ would be even less than that given above. The factor \/R/fE in equation ( 1 ) is derived upon the assumption that only the crystal losses are significant. If this is not to be the case, this factor should be replaced by one equal to / total energy expended per cycle V energy stored ‘ ’ though it would seem from equations (4) and (5) that Rg cannot be safely made more than 5 to 10 times larger than the minimum PI, particularly if an AT cut is employed, since it has a value of r of only 250, and since Cg is normally no greater than 4 or 5 times Cm„, it should be remembered that the value of km above is based upon the assumption that no energy is being fed to the circuit during an entire cycle, so that the net loss is equal to the gross loss. This condition is only approached in high-efficiency class-C circuits where the operat- ing bias is several times the cutoff bias. Except in the case of power oscillators, such operation is not feasible because of the high operating voltages that are required. The larger the fraction of the cycle during which the tube is conducting, the larger the ratio of the usable Rg to that given by equation (3) . If the tube is conducting one-half the time, class-B operation, the maximum safe R, is more than twice that given by equation (3). For class-B and class-C operation, the output stability is almost entirely dependent upon the automatic adjustment of the bias, for any decrease in signal strength will mean a decrease in over-all amplifi- cation unless the bias can drop immediately to allow more energy to be fed to the circuit. On the other hand, it was found in paragraph 1-294 that if an oscillator tube is operated at a bias imme- diately above the knee of its E.-L curve, the bias can remain fixed and the variations in excitation directly produce a change in amplification that tends to annul the original variation. If the tube WADC TR 56-156 146 voltages are so selected that a gridleak bias at equilibrium is also at the optimum fixed-bias point, then limiting can be achieved both from the grid- leak action and the excitation swings. Under these conditions, R* can be safely increased to values beyond one megohm, even at high frequencies. As a design consideration, however, the gap between the theoretical and the practical solution can prove quite wide. Among the optimum-bias bugs that resist extermination; there is the difficulty of find- ing a vacuum tube having the desired operating characteristics, and once found, there is the addi- tional problem of maintaining an optimum oper- ating state with crystal units having different values of effective resistance. These problems are discussed in some detail in succeeding paragraphs. The main problem is to reduce the grid losses to negligible proportions without endangering the output voltage stability. This can normally be done with any parallel-resonant crystal oscillator if the tube is conducting throughout most of each cycle. Gridleak Resistance and Output Control in Pierce Circuit 1-301. If it is necessary for a Pierce oscillator to provide a higher voltage output than can be ob- tained under the conditions of maximum fre- quency stability, the C^/C„ ratio can be increased. If the total load capacitance is to remain constant, Cg will necessarily be larger, and the excitation voltage smaller, so a smaller value of R, can be used without the grid losses becoming significant. If the capacitance ratio is to be adjustable in order to permit an operator or technician to control the output voltage, R^ cannot he made larger than that value which would permit a stable output with the largest operable value of Cg at the highest fre- quency at which the oscillator is to be used. If such an adjustment is to be provided in a Pierce circuit, Cg and C„ should be so ganged as to always provide a constant load capacitance. This problem is dis- cussed in paragraph 1-318. Insofar as the grid-to- cathode resistance is concerned, the maximum safe value of R, becomes less if Cg is to be variable than otherwise. Without changing the Cg/C„ ratio, larger outputs can be achieved by reducing the value of Rg to a point where the grid leakage is so great that the average bias is considerably smaller than the peak excitation voltage. With this the case, the oscillations must build up to higher amp- litude levels before equilibrium can be reached. Al- though the maximum excitation is still fixed by the rated drive level of the crystal unit, the output can be controlled somewhat within this restriction by a variable R,. At the higher frequencies, this Section I Crystal Oscillators method of output adjustment requires such low values of R* that the grid losses seriously affect the frequency stability. However, at very low fre- quencies, a variable R^ could be feasible as a means of adjusting the output of a Pierce circuit to a desired level when one crystal unit is replaced by another of different effective resistance. Although such a design feature has no particular recom- mendation, it could be preferred over those meth- ods of output control that require adjustment of the Cg/Cp ratio, which risk changes being made in the total load capacitance. With grid control, the lowest adjusted value of R^ could be designed to provide the desired output when a crystal unit of maximum effective resistance (minimum PI) is connected in the circuit; whereas the larger values of R, could ensure the same output with some theoretical minimum value of effective resistance. Since the crystal current, U, is practically constant as long as the output voltage E,, is constant, the power losses in the crystal, equal to U-'R,., tend to vary directly with R„ as long as E„ is held constant by adjustments of Rp. Under those conditions where the capacitance ratio does not change, a maximum crystal drive level is required for the crystal unit of maximum R,., and a minimum crys- tal drive level when R,. is a minimum — the reverse of those conditions discussed in paragraph 1-294 when a fixed bias instead of a fixed output is assumed. 1-302. As applied to crystal oscillators in general it cannot be said that a variable gridh'ak resistance is advisable except for test purposes or unless its purpose is to obtain the minimum possible grid losses when changing from one crystal unit to another. As an output-voltage control device other methods are generally to be preferred. Except at very low frequencies, the resistance values neces- sary to appreciably lower the average bias are too small to prevent the grid losses from becoming a significant frequency-determining factor. This statement, of course, only expresses a general rule, and in specific instances the inter-relations among the circuit variables may be such as to annul the effects upon the frequency. For example, the bot- tom set of curves in figure 1-129 (A) is to be ex- pected theoretically to indicate a greater frequency stability when R^ is 1 megohm rather than when it is 0.2 megohm, but this effect was not observed. Figure 1-130 shows curves of output voltage ob- tained from the same experimental oscillator that was used in plotting the curves of figure 1-129. Although the curves are plotted as output-voltage versus crystal driving power, it should be under- stood that the actual independent variable for each WADC TR 56-156 147 Section I Crystal Oscillators curve is the plate voltage. Each curve represents a particular value of grid resistance. The cross lines intersect the curves at points corresponding to the same values of plate voltage. From figure 1-130 it can be seen that large percentage changes in the grid resistance can cause changes in the output voltage on the order of 30 per cent or so, but which increase sharply as becomes small. 1-303. If an ad j usable output voltage is desired, probably the best solution to the problem is to use a screen-grid tube having an r-f-bypassed, varia- ble, voltage-dropping resistor in series with the screen supply voltage. Varying this resistance will control the output of the tube and the crystal driv- ing power. The maximum permissible output volt- age must be determined on the assumption that the crystal unit has the maximum permissible R,. E Since Ig «= y — maximum permissible Ig = V^cm/Rem, wherc P,.m and Rem are the maxi- mum crystal driving power and effective resist- ance, respectively, then the maximum permissible constant Ep is (max) Ep = (X, -f Xc,) V Pom/R,™ 1—303 (1) X If it is assumed that Xc, Xop » , where X,(= — l/mCi) is the total load reactance equal and opposite to X,, equation (1) becomes (max) Ep = 2a>C, 1—303 (2) According to equation (2), the maximum permissi- ble constant Ep varies inversely with the crystal frequency. If the oscillator is to be used at more than one frequency, and at the same time is to provide the same output voltage regardless of the frequency, the maximum Ep is that value given by equation (2) for the crystal unit of highest fre- quency, assuming the crystal specifications are the same for all frequencies. With the ratio of Cg/Cp approximately equal to one, equation (2) also gives the value of Eg, which obviously will also remain constant. R, can be made quite large, so that |Ec| will approximately equal E„\/2^. With Ec constant, and with the plate voltage Ep also assumed to be constant, the operating position of the tube on the Eolp curve largely becomes the function of the screen voltage. As the screen voltage is increased, g„, increases, which means that the slope of the CRYSTAL DRIVE- MILLIWATTS figure I-I30. Output curves of tuned Pierce oscillator for different values of grid resistance when reactance of tuned plate circuit is adjusted for output voltages equal to 50 percent of the maximum attainable. A 7-ific CR-I8/U crystal unit was used, having a PI of 49,000 ohms when operating into its rated load capacitance of 32 /i/J WADC TR 56-156 148 $«cHon I Crystal Oscillators EJb curve becomes steeper. Also, the cutoff bias is increased. Since E, is being held constant, the effect is one of shifting the operating bias, percent- age-wise, closer to or farther up the straight por- tion of the EJi, curve. From the point of view of using as large a value of Kg as is possible, it is desirable that the operating position be just above the knee of the EJb curve when the screen voltage is to be a maximum, i.e., when R, of the highest- frequency crystal unit is a maximum. Gridleak Resistance and Oscillator Keying of Pierce Circuit 1-304. If avoidable, a crystal oscillator should not, itself, be keyed. For one reason, the oscillation build-up time is not negligible if rapid telegraph keying is desired. As the operable speed limit is approached the wave shape becomes distorted and the harmonic output is considerably increased. Even the keying of a crystal oscillator in a push- to-talk voice transmitter is not desirable if fre- quency stability is important, since on-and-off operation constantly raises and lowers the crystal temperature. Thus, the frequency is kept in a state of constant variation to a degree dependent upon the magnitude of frequency-temperature coeffi- cient of the crystal unit at the average operating temperature. Unless necessary for reasons of econ- omy in space, cost, or the like, the oscillator should be designed for continuous operation and the key- ing performed in one or more of the succeeding amplifier stages. Usually the keying circuit is de- signed to remove and apply by one means or another, a cutoff bias in the buffer-amplifier stage. During the time that the buffer amplifier is cut off, the crystal circuit continues to oscillate, but the signal cannot be amplified and applied to the suc- ceeding stages. 1-305. When it is necessary to key the oscillator, itself, the reason is normally that the space and weight requirements are so limited that no more than one or two vacuum-tube stages can be al- lowed. For this same reason, the oscillator is prob- ably required to develop as much output power as possible, so that a Miller, rather than a Pierce cir- cuit is generally employed if crystal control of the frequency is required. Nevertheless, the factors affecting the build-up time are approximately the same in either circuit. Fundamentally, the reason that a crystal oscillator requires a relatively much longer build-up time than does a conventional in- ductor-capacitor tuned circuit of the same reso- WAOC TR 56-156 149 Section I Crystal Oscillators nant frequency, is because the energy to be stored in the crystal is much greater than that which would be stored in an inductor-capacitor circuit. For a given tank current, the stored energy is proportional to the inductance, so, to a first approximation, we can suppose that the build-up time of, say, a Pierce oscillator as compared with that of a Colpitts oscillator of the same frequency, is directly proportional to the inductance ratio. On the other hand, it can be imagined that the build-up time tends to vary inversely with the total effective resistance in the tank circuit. The greater this resistance, the more quickly do the losses in the circuit rise to equilibrium with the rate of en- ergy supply. The build-up time also tends to vary inversely with the frequency. Clearly, if the fre- quency were one cycle per second, equilibrium could not be reached in a shorter period. Finally, the build-up time is a function of the electro- mechanical coupling of the crystal to the circuit. The larger the C„/C ratio of the crystal unit, the weaker is the coupling and the longer is the period before equilibrium can be reached. The exact rela- tions of all the circuit variables in an equation expressing the time required for the amplitude to rise to within one per cent or so of its equilibrium limit would, indeed, be quite involved. Insofar as the crystal is concerned, the build-up time can be expected to vary positively if plotted against L, C, Co, and 1/R of the crystal unit. The percentage variation of the build-up time with a given per- centage variation in L can be expected to be greater than with the same percentage variation in C, because of the fact that, say, an increase in C, although increasing the build-up time by lowering the frequency, will also tend to decrease the build- up time by improving the electromechanical ratio. Thus, if the frequency remains constant, a crystal oscillator can be keyed at a faster rate if the L/C ratio is kept to a minimum, provided C„ is not in- creased. In other words, a crystal element should be chosen that has as large a piezoelectric effect as possible, provided the frequency-temperature co- efficient is small. For example, for high-frequency circuits, an AT-cut crystal which has a capacitance (] ratio 250 is to be expected to provide better keying characteristics than a BT-cut crystal, which has capacitance ratio of 650. Preferably, from the point of view of a maximum keying speed for a given output voltage, the gridleak resistance should be kept small, not only to load the circuit and to provide quick-action limiting, but also to keep the positive swings of the grid and the trans- conductance high. The oscillator will almost cer- tainly be designed for maximum power output, so that the tank circuit will be well loaded, for which reason the grid resistance must be kept relatively small as a safeguard against intermittent oscilla- tions. It is questionable as to just how much the effective tank resistance limits the build-up time. Of course, if the resistance were zero, the oscilla- tions would theoretically continue to rise indefi- nitely. On the other hand, the time required for the amplitude to reach any given value is least when the energy being lost from the circuit is least. In this respect, the build-up time tends to vary directly, not inversely, with the tank resist- ance. It would seem, that to obtain a maximum keying speed it might be preferable to use a fixed bias or a cathode bias, instead of the gridleak ac- tion. Using a sharp-cutoff tube biased for class-A operation, a grid, plate, or output circuit limiting arrangement could permit the oscillations to build up to a given level under conditions of a maximum ratio of input to dissipated power. Above this amp- litude level the ratio would sharply decrease. Such a circuit could raise the permissible keying speed, but since this is accomplished by virtue of sudden changes in the circuit parameters, which changes always accompany to some extent any limiting ac- tion, an increased frequency instability and har- monic output are almost certain to result. Although a crystal oscillator should not be designed to be keyed unless absolutely necessary, experimental circuits have obtained keying speeds approaching 400 words per minute. The higher the keying speed, however, the greater must be the frequency tolerance. Gridleak Resistance When Pierce Circuit Permits Switching from Crystal to Variable LC Control 1-306. It is often necessary to provide a variable- tuned, inductor-capacitor auxiliary circuit to per- mit emergency operation at frequencies other than those provided by the available crystals, or in the event of crystal failure. For this purpose it is often possible and is usually desirable to use the same vacuum tube that is used during crystal control. For example, a Pierce circuit could be readily con- verted to a Colpitts circuit simply by switching from the crystal to a tuning inductor, or to an inductor shunted by a variable capacitor. However, when such a conversion is made, the ratio of the stored energy to the power dissipation becomes much smaller than that during crystal control. For this reason, the maximum safe value of gridleak resistance is much smaller than during crystal operation. For output voltages comparable to those obtained with crystal control, the LC circuit em- WADC TR 56-156 150 ploys gridleak resistances ranging ^011/20, 000 to occasionally 100,000 ohms. If the LC ydrcuit is in- tended to furnish a much greater output than the crystal circuit, lower values of lymay be neces- sary. Rather than require the^ystal circuit to operate with small values of R*y^t would be prefer- able to connect an additional ainunt resistor in the grid circuit when switching to variable-tuning control. Gridleak Resistance When Used with Cathode Biasing Resistor in Pierce Circuit 1-307. In addition to the voltage across the grid- leak resistance, part of the bias voltage can be furnished by an r-f-bypassed resistance in the cathode circuit. The cathode resistor protects the tube from excessive plate current should oscilla- tions cease, and has the additional advantage of reducing the grid current and, hence, the grid losses. The power expended in the grid circuit will be approximately equal to where Et is the total bias and I,, is the grid current. Actually, un- less an r-f choke is used in the grid circuit, the grid losses will be somewhat greater than E^Ic because of the a-c component of voltage across Rj. As the cathode component of the bias becomes small, the grid losses approach 1.5 as a limit. See para- graph 1-296. The values of the cathode resistance, Rk, usually range from 100 to 1000 ohms. The re- actance of the bypass capacitor should be at least as small as Rk/10 at the lowest operating fre- quency. With Rk connected between cathode and ground, the d-c voltage developed equals ( Ib -f- 1. ) Rk ; or approximately, ERk. The total bias. Saction I Crystal Oscillators E„ is still approximately equal to V2 E,. The d-c grid current is given by the equation I E, I - Ek V2 E, - E Rk R. R. 1—307 (1) where Ek is the voltage across the cathode resistor. AGC USED WITH PIERCE OSCILLATOR 1-308. Where space and cost permit, optimum out- put stability can be had when the oscillator bias is provided through an automatic-gain-control cir- cuit. Gridleak action can be effective in initiating oscillations, but the bias furnished through AGC should be of much greater magnitude in order to be of maximum effectiveness. A small increase in output voltage must cause a large increase in bias. The use of AGC reduces the grid losses to a mini- mum and maintains a constant amplitude of os- cillation. It is this latter feature that is, of course, of most importance — particularly so when the same oscillator is to be switched from one crystal unit to another. The voltage requirements for con- stant output without risking the overdrive' of any of the crystals are the same as those that apply in the case of manual adjustment of the output. (See paragraph 1-302.) An A-G-C circuit applicable for use with a Pierce, or Miller, type oscillator, is shown in figure 1-131. The oscillator output is amp- lified by Vj. The output of V2 is then rectified by V:,. The oscillator bias equals the average rectified voltage across R3. C, bypasses the r-f component to ground. If R, were increased indefinitely the WADC TR 56-156 Figure I-I3T. Pierce osciilator with automatic gain control 151 Section I Crystal Oscillators bias voltage would approach in magnitude the peak value of the Vj output voltage. Ri, Rj, and R, not critical — each can be made equal to 50K/U a faster-acting gain control is required. However, R, should be kept as large as possible. Ass^e that the r-f losses in R,, R,, and Rj are negl^ble and that C,, and Cg are approximately equal, 4o that V, is operating into a load impedance ai^roximately equal to PI/4. Under these conditions gm will be the minimum and Rp the maximum possible for sustained oscillations as long as the load capaci- tance across the crystal unit remains constant. The actual values of g„, and R„ are fixed by the vacuum- tube characteristics. Although the effective pa- rameters of the tube are directly dependent upon the peak-to-peak magnitude of the excitation, as well as indirectly through the bias, it can be said that to a first approximation the equilibrium Rp and g,„ are associated with a bias of more or less definite magnitude if the plate voltage is constant, and that approximately the same bias must exist regardless of whether it is developed by gridleak action or by AGC. Thus, the difference between AGC and gridleak control is not primarily in the magnitude of the bias, but in the amplitude of os- cillations. Gridleak action requires that the peak excitation voltage of Vj be slightly greater than the required bias ; AGC requires that the peak ex- citation voltage of V 2 times the voltage amplifica- tion of the V 2 stage be slightly greater than the required bias of V,. If the peak excitation voltage of Vj is assumed to equal Ep^, which, in turn, is assumed to equal 1^, and k 2 is the effec- tive amplification of tne V, stage, then I E, I = kj E,™ 1-308 (1) or Egm = I Ec/kj I Since E,. is approximately fixed, it can be seen that the amplitude of oscillations is only ^ as large as those that would exist by tlie gridleak method em- ploying the same plate voltage. This is not a de- sirable feature where large output is required, but from the point of view of ensuring a low crystal drive and maximum stability, an A-G-C circuit has great advantages. Although the equilibrium amp- litude is low, oscillations start as readily as with gridleak bias. AGC permits class-A operation with remote-cutoff tubes, and, since the limiting is very slow-acting, very pure sine-wave outputs and ex- cellent frequency stability as well as amplitude stability is obtainable. PLATE-SUPPLY CIRCUIT OF PIERCE OSCILLATOR 1-309. For optimum frequency stability it is im- portant that the r-f impedance of the B+ circuit be as high as possible relative to the impedance of the tank. If Cg/Cp = 1, the tank impedance equals PI/4. If the oscillator is intended to oscillate at only one frequency, or within a narrow fre- quency, range, it is generally preferable that the B+ voltage be fed through an r-f choke. This method affords a maximum impedance with mini- mum loss and minimum voltage at the B+ source. The inconvenience of an r-f choke is that its im- pedance changes with frequency, being inductive below its effective parallel-resonant point, and ca- pacitive above. As long as this effect does not change the effective value of C„ by more than ±10 per cent, the total load capacitance will not change by more than 5 per cent, if Cg/Cp = 1. Within these limits the use of a choke is to be preferred. For wide frequency ranges, a resistor should be used in the plate circuit, such as R, in figure 1-131. It is desirable for this resistance to be as high as 60K, or higher, from the point of view of frequency stability. On the other hand, the larger the resist- ance the higher the B+ voltage source must be to provide a given plate voltage. Plate-supply resist- ances on the order of 5000 to 10,000 ohms have one other important advantage besides permitting lower B+ sources. They load the oscillator tank so that differences in the resistance of the crystal from one unit to the next have very little effect upon the output impedance of the tube. Hence, . when a change is made from one crystal unit to the next, the output voltage remains approximately the same. 1-310. The proper compromise in selecting a plate- circuit resistance depends upon the frequency-tol- erance limits. The plate-circuit resistance does afford a certain frequency-stabilizing effect that is not provided by an r-f choke, particularly so when age is used. The effect is one of reducing the change in Rp of the vacuum tube caused by a change in grid bias. For example, if the bias be- comes more negative Rp increases, and Ib, the aver- age plate current, decreases. There is then less voltage drop across the plate-supply resistor, and the resulting increase in plate voltage tends to de- crease Rp, thereby annulling part of the increase in Rp due to the change in bias. The plate-voltage source should be regulated, if good stability is re- quired. Where the frequency deviation must be kept to a minimum, the oscillator may require a separate rectifier unit, filter circuit, and voltage- regulator circuit. WADC TR 56-T56 152 Saction I Crystal Oscillators CHOOSING A VACUUM TUBE FOB THE PIERCE CIRCUIT 1-311. It is no problem to find a vacuum tube that will permit a Pierce circuit to oscillate. Indeed, one of the major problems in tube circuit design is to prevent oscillations from occurring. With a crystal connected between the plate and grid of any vac- uum-tube amplifier, the stray capacitance in the circuit is usually sufficient to cause oscillations to build up. If the plate voltage is not so high that the crystal is over-driven, the frequency stability of a stray-capacitance circuit may even be satis- factory for general-purpose use. Thus, the problem is not to find a vacuum tube that will work, but one that will be most satisfactory from the point of view of output stability and cost. First, a large tube is not necessary, since the Pierce circuit is not suited for large output. The choice of tube will depend somewhat upon the exact purpose of the oscillator and of the equipment of which it is a component. If the frequency tolerance is to be large, little thought need be given to fine points in the design, for the principal problem will be to keep the production costs to a minimum. A triode would be satisfactory, a 5K to 50K resistance in the plate circuit, a Cg/Cp ratio between 1 and 2, and a plate voltage sufficiently low so that the driv- ing power of the crystal does not exceed the rated level for any effective crystal resistance meeting the specifications. A high-mu triode generally pro- vides the better frequency stability because of its larger effective Rp, but it will have a higher plate dissipation for the same output voltage. Of the high-mu triodes, probably the 6AB4 is to be pre- ferred as a simple unit, and the 12AX7 and the 12AT7 as twin triodes. It is the medium-mu tube that has been the most favored by design engi- neers when a triode has been chosen. Of these the 6C4, 6J4, and 7A4 single units, and the 6SN7-GTA twin unit are among the more popular. The 6C4 and the 6J4 are to be preferred for high-frequency operation. Since the 7A4 and the 6SN7-GTA have approximately 4 iijd capacitance between grid and plate, the 6C4, 6J4, 6.16, or the 12AU7, each with 1.5 nnf capacitance grid to plate, should provide the better frequency stability — particularly at high frequencies. The 7A4 and the 6SN7-GTA are generally more satisfactory for use in a Miller cir- cuit. Where greater frequency stability is required, a pentode should be used. A pentode has the ad- vantages of low plate-to-grid capacitance, greater Rp, and a screen grid whose voltage can be ad- justed independently of the control-grid bias and plate voltage, thereby permitting a greater range of adjustments in the plate characteristics. Con- ventional pentodes must be operated at reduced voltages, to avoid overdriving the crystal, unless rather high Cg/Cp ratios are used. Subminiature pentodes have operating characteristics at their normal operating voltages ideally suited for crys- tal drive levels. The 1U4 is one such type having a sharp cutoff. Among the miniature pentodes hav- ing a sharp cutoff, the 6AU6, 6BC5, and 6AH6 are tubes generally recommended for wide-band, h-f circuits. The 6CB6, although designed principally for television use at 40 me, should also be quite appropriate in crystal oscillator circuits. Remote- cutoff tubes are generally used only in special cir- cuits. For example, if a low harmonic output is required, such tubes could be employed in conjunc- tion with AGC. When the harmonic content is not of first importance, AGC is more effective if used with sharp-cutoff tubes, where a slight change in grid bias can make a much larger change in small- signal outputs than is possible if the slope of the E..I|, curve changes very gradually. Actually, re- mote-cutoff tubes, when used, are usually found in doubler circuits, because of the large second-har- monic component that is produced. Although class-B and class-C operation with sharp-cutoff tubes can produce even greater harmonic outputs, there is the problem of ensuring that a crystal of large R,. will not be overdriven if it is to be oper- ated in a class-B or class-C circuit. The output voltages of remote-cutoff tubes tend to vary more with crystals of different resistances than is the case when sharp-cutoff tubes are used. The reason is that in the former case the effective I,, continues to increase as R, becomes small, since very large excitation voltages are required to override the cutoff point. On the other hand, the effective Ip begins to decrease when class-B operation is ap- proached and such operation can be had with rela- tively small excitation voltages when sharp-cutoff tubes are used. If a remote-cutoff tube is desired, recommended types are the subminiature 1T4, the miniature 6BA6 and 12BA6, the lock-in 7A7, and the conventional-sized tubes such as the 6SK7 and 12SK7. The mention of particular vacuum tubes here should not be construed as official recom- mendation; they are named simply because they are the tubes commonly found in new equipment. The design engineer may very well find that the characteristics of other tubes are more appropri- ate for his needs. Pierce-0 scillator Design Considerations When Vacuum Tube with Very Sharp Cutoff is Used 1-312. In making a preliminary approximation as to what the performance of a particular tube will WADC TR 56-156 153 Section I Crystal Oscillators be if used in a Pierce circuit, it should first be kept in mind that the ratio ”■ Cg/Cp mZl Rp + Zl gm Rp Zx, Rp + Zl 1—312 (1) is the gain of the tube. If the gain = k, and if R — 2 . is 10 or greater, then Zl gm k/Z,. 1—312 (2) or Rp = MZL/k 1—312 (3) Either equation (2) or (3) can be used to esti- mate approximately the grid bias for a given plate voltage, and vice versa, that can be expected if a particular tube is used. Assume, for example, that k = 1 , that gridleak bias is to be used, and that the grid and load losses are negligible compared with the crystal driving power. In this case, the minimum expected Zl will equal (min) PI/4, which occurs when a crystal unit has the maximum allowable and is operated at the rated load ca- pacitance, Cj. Under these conditions, the maxi- mum permissible bias, as given by equation 1-293 ( 2 ), is (max) Ee = - V 2 Pen, / 2a)C, V R,„, This maximum value of E,. is to be interpreted as a maximum that can be allowed only if R„ is a maximum or if the output voltage is to be the same magnitude regardless of the value of R,. In this latter case, P,.„, and R,.„, fix the output and bias limits for all crystal units of a given type. The constant output can be obtained in several ways: by the use of an actual or equivalent, parallel, plate load resistance that is small compared with the minimum nonloaded crystal tank impedance; by the use of AGC, by the use of manual voltage ad- j ustments ; or by other methods. The present dis- cussion concerns only the noncontrolled nonloaded circuit. If a crystal unit of maximum R„ being driven at the maximum drive level, is replaced by a crystal unit of smaller R„ Zl increases, and Ep and Ig tend to increase proportionately, so that the crys- tal driving power, equal to 1 *= Re, is greater than when Re is a maximum. According to equation (3) , insofar as it can be assumed that /* remains ap- proximately constant (in practice, decreases somewhat) R,. increases proportionately with Zl, so that although the equivalent generator voltage, —y-Eg, increases, I,, remains constant. Thus, Ip => g,„Eg kEg/ZL constant. In an actual circuit where Rp > > Zl and the vacuum tube has a very sharp cutoff, the effective Ip increases up to the point that the tube is cutoff for approximately three-fifths of the negative alternation (three- tenths of the entire cycle) . As the excitation volt- age increases beyond that point. Ip progressively decreases, although the total power supplied to the tank circuit continues to increase as long as the excitation voltage continues to increase. The con- clusions above are derived in the special case of a Cg/Cp ratio of unity, by assuming that for all prac- tical purposes the plate-current pulses are in phase with Ep, and that Ep > > Ep. Figure 1-132 illus- trates different states of operation of the same oscillator circuit that can occur if crystal units of the same frequency but different values of Rp are inserted in the circuit. A change from the class-A to the class-C state could readily occur if the crys- tal Rp were reduced by more than one-half. The effective Ipm is defined by the equation P 7 .L = Ipn. Ep,„/2 1-312 (4) where Pz,. is the power expended in the tank cir- cuit. Since Z,, is very small compared with Rp, it can be assumed that the sinusoidal component, Ep, of the with-signal d-c plate voltage, ei„ is negligi- ble by comparison with the average value, Ep ; that is, Ep ±: Ep„, «= Ep. With this assumption we can treat Ip„„ the value of the with-signal, d-c plate current, at the positive peak of excitation (e,. = 0 ) as a constant. The assumptions above also imply that very little grid current exists ; otherwise, the larger excitation voltages would drive the grid considerably above zero at the positive peaks. With the peak instantaneous d-c plate current a con- stant, the total energy supplied by the power source progressively decreases as Zl and the ex- citation increase, since Ip, the average ip, becomes progressively smaller, whereas Ep remains con- stant. (Actually, if the plate current is supplied through a resistor, a decrease in Ip causes Ep to increase somewhat. For the problem at hand, assume that a regulated B+ is fed through an r-f choke.) Thus, it can be seen that as Rp becomes small the plate efficiency increases considerably. However, the efficiency of a crystal oscillator does not approach the high ratios of input to tank power that are obtained with conventional class-C power amplifiers and oscillators. The latter cir- cuits can operate at efficiencies of 60 to 90 per cent because approaches Ep in magnitude. The in- stantaneous power being dissipated in the tube is the instantaneous value of ipOp, and the instanta- neous power being delivered to the tank is ipep. When ip is a maximum, ep = Ep — Epm < < e^ = WADC TR 56-156 154 S«cHon I Crystal Oscillators E,kb, so that most of power goes to the tank circuit. In the conventional Pierce oscillator such high efficiency is not to be approached unless the C*/Cp ratio is to be made very large and Eb approaches in magnitude the voltage specifications of the crys- tal unit. Now, to obtain a maximum output with- out the risk of overdriving a randomly selected crystal unit, it will be useful to derive approxi- mate equations concerning the change in crystal power with a change in R*. The crystal power, we shall assume to equal the total tank power, Pzl. In short, the problem is to be able to express Pz, as a function of R,. Ibm, Et, and E™ (the cutoff voltage) will be considered constants, and ip and ep are to be assumed to be in phase. First, we express the effective I,„„ for each class of operation in terms of the constants above and the angles <t) and 6, where appropriate. (See figure 1-132.) As a safeguard against intermittent oscillations, which are most likely to occur when R^ is a maximum, assume that the bias for maximum R^ is to occur on the straight portion of the Eelb curve. If the oscillations are to build up at all, they must con- tinue to do so until the negative excitation peak at least extends into the lower bend of the EcIb curve, for it is only beyond the straight portion of the curve that g„ can change in order to seek its equilibrium value — that is, unless R* is so small that equilibrium is reached by virtue of the in- crease in grid losses alone. With a large Rg and a reasonably sharp cutoff, it is virtually impossible for oscillations to start if the amplification is not at least sufficient to increase the excitation to where the negative peak is very nearly equal to E„. Assume, then, that with R, — Rem, the oscil- lator is designed to operate approximately as shown in figure 1-132 (A). It can be seen intui- tively that (Class A) Ipm - Ibn./2 1-312 (5) and with C,/C„ = 1, considering only the unsigned magnitudes of the bias voltage, (Class A) Epm = Ee = Ee„/2 1-312 (6) LARGE Rg — — K SMALL Rg Fi'Sure I-I32. Change of state of Pierce oscillator with a Cp/Cp ratio of one, under no-load conditions when Cl, is held constant and the effective resistance of the crystal changes WADC TR 56-156 155 section I Crystal Oscillators so that (Class A) Pzl = Ip„ Ep„/2 = Ibm Eeo /8 1—312 (7) Ecolbn, /*' + 2T(l + 8in«)y^ ♦ (sin^ wt + sin 4> sin o>t) doit 1—812 (16) Equation (7) represents the maximum possible crystal power if a tube is not to be driven beyond cutoff. Referring now to figure 1-132 (B), we shall assume that a crystal unit with an R. slightly less than the maximum is connected so that the bias is similar to that under AB operating conditions. In, represents the apparent maximum Ip. It can be seen that except for the angle (w — 2 </>), when the tube is cut off, ib = In, (sin cot -|- sin 0 ) 1 — 312 ( 8 ) where <t> can be considered a constant. Now, Ibm = Im (1 + sin 0 ) 1 — 312 (9) so, on substitution in equation ( 8 ), ib = , , (sin cot -I- sin 0 ) 1—312 (10) 1 "T Sin <t> Similarly, Eco = Eg,n (1 -f sin 0 ) = Ep„, (1 - 1 - sin 0 ) 1—312 (11) so that On integration. (Class AB) Pzl = Eco Ibm (x + 2 0 + 2 sin 2 0 ) 4t (1 -|- sin 0 )* 1—312 (16) No maximum exists for equation (16) with values of 0 between 0 and ir/2. For class-A operation similar to that in figure 1-132 (A), 0 = ir/2, so that equation (16) becomes (Class A) Pzp = E.. Ibm /8 This checks, as is to be expected, with equation (7) . For class-B operation, 0 = 0, so that equation (16) becomes (Class B) Pz. = E,„ Ibm/4 1—312 (17) Note that the power expenditure in the crystal unit for class-B operation is exactly twice that found for class-A operation. Since Epm under class-B conditions is equal to Ec„ (see figure 1-132 (C)), or twice the class-A value of Epm, then, be- cause Pzt = IpmEpm/2, the effective Ipm at class B must be equal to the same effective value as at class A. Thus, Cp = Epm sin cut E CO 1 -t- sin 0 (sin cut) 1—312 (12) Since no energy is being supplied during the time that the tube is cut off, the energy delivered to the tank per cycle is ^ (»■ + «) Cp ib dt (- 0 ) if- IT + Cp ib dcut 1—312 (13) where t = time in seconds. Thus, the energy delivered per second, is P Zt ib dcut I Cp ib dcut 1—312 (14) On substitution of Ep and Ib from equations (10) and ( 12 ), WAOC TR 56-156 (Class B) Ipm = Ibm/2 1-312 (18) Also, since Epm = IpmZi., if Epm has doubled but Ipm has not changed it can only mean that Zl has doubled. In other words, if the oscillator is de- signed to operate class A when Re is a maximum, it will operate class B when a crystal unit is in- serted that has an effective resistance equal to Rem/2. Equation (16) can be generalized to apply for all operating states in which 2 £ma is equal to or greater than E<.o. For greater simplicity, 0 should be replaced by («■ — $) / 2 , where = *■ — 20 is the angle during which the tube is cut off. $ is always positive, whereas 0 would be negative in the case of class-C operation. Thus, equation (16) can be expressed (all classes) Pzt = Eco Ibm [2ir — e + sin d) 4x^1 -t- cos 1—312 (19) 156 The slope of this equation is positive for all values of B less than 2*, so that the power dissipated in a crystal unit always becomes greater as R, be- comes smaller. By substituting cos for sin ^ in equation (11) and rearranging, we have Ep„ = E„/(i + co 8|) so that 2 P Ipn, = = 2 Pz. I 1 + COS "nm (■ Ibm (2ir — -f sin 0) 2t(i + cos|) 1 )/^ 1—312 (20) Equation (20), unlike equation (19), has a maxi- mum when 6 is approximately 3jr/5. That a maxi- mum (or a minimum) occurs between 8 = 0 and 0 = IT is to be expected, since Ip™ has the same value for each of those values of B. This maxi- mum is (max) Ip™ = 0.54 Ib™ 1-^12 (21) Now, Z,, = 2 Pz../ (Ip™)* 1—312 (22) On substituting equations (10) and (20) in (22) Section I Crystal Oscillators which could have been predicted on the basis of equation (2). Assume that equation (25) holds, what will be the value of 6 when a crystal unit having a practical minimum value of R. equal to Re™/9 is connected in the circuit? When R, = R„, the negative term within the parentheses of equa- tion (24) is equal to —1; with R, = Rem/9, the same term is reduced to —1/9. Thus, the maximum 0 to be expected is defined by : ((max) 0 for (min) R,] when: 9 — sin 0 = 16ir/9 1—312 (26) Figure 1-133 shows that equation (26) requires that (max) 0 = 16t/9 - 1 1—312 (27) In other words, when a sharp-cutoff tube is used and the oscillator is designed for class-A operation with crystal units of maximum R,, the oscillator will be operating class C, with the tube cut off approximately three-fourths of the time, when crystal units of minimum values of R, are con- nected in the circuit. Equation (24) can be gen- eralized to define 6 with reference to any conven- ient value of R,, simply by assuming that 6 = 0 when R, = (ref) R,. Thus, 9 - sin i9 = 2ir [1 - R„/(ref)Re] 1—312 (28) or Zl = 2tE. Ibm (2 t — 9-1- sin 9) 1—312 (23) Rearranging and substituting l/4<»’C,’Rp for Z,,, where C, is the specified load capacitance of the crystal unit. 9 — sin 9 = 2ir (1 — 4u*C, Eco R./Ib™) 1—312 (24) Equation (24) is quite significant in that it pre- dicts the approximate angle during which a given tube will be cut off for a given value of R,. A Pierce oscillator designed so that the tube is oper- ating with a class-A bias equal to Eco/2 when R, is a maximum will have a value of 9 equal to zero. Thus, when R, = R™, each side of equation (24) must vanish. For the right-hand side to equal zero, 4a)* C, Eoo Rem = Ibm This is equivalent to saying that 1 ^ 4 (min) Zl (min) PI = (average) g™ ii^co 1—312 (25) 9n - sin 9 n = 2ir (N - 1)/N 1—312 (29) where N = (ref) R./Rej,, 9 = 0 when R, = (ref) Re, and 9 n is the value of 9 for the particular value of Re symbolized by Re,,. The reference R* need not be the maximum permissible R,. For a given oscillator of Cg/Cp ratio equal to 1, (ref) R, would be the value of Re that would cause the peak-to- peak excitation voltage to equal Eeo in magnitude. Assuming that (ref) R, = Rem, what then will be the ratios of Pz,, and I,™, corresponding to mini- mum and maximum values of R*? When 9 = — , as given by equation (27), cos 9/2 is very nearly —2/3, so equation (19) becomes (Class C max) Pz^ _ ~ ~9~) _ Eeo Ibm 4x(l-|) ^ 1—312 (30) On comparison with equations (7) and (17), which give values of Pz,. of EeoIbm/8 and EeoIbm/4 WADC TR 56-156 157 PIERCE OSCILLATOR OPERATION AS A FUNCTION OF CRYSTAL Rf. 0 • ANGLE DURING WHICH TU8E IS CUTOUT. /under conditions of no load , sharp-cutoff TUBE,\ IF; 8 ■ 0, WHEN crystal R, IS MAX. PERMITTED, UaRGE Ro.AND Co/Cp*t I then: e«-e, when R, - (MAX) R,/N; Saction I Crystal Oscillators NIS WADC TR 56-156 158 Figure 1-133. Sine curve plotted to tame scale as the angle B. 45-degree diagonal that intercepts sin Bi- on curve intercepts (dk—sin dj on 8 axis SacKon I Crystal Oscillators for clasa-A and class-B operation, respectively, we find that where there is to be no output control, the no-load tube voltage must be so chosen that a crystal unit of maximum R, is not driven at more than one-fourth the rated drive level, otherwise crystals of small R, will be overdriven. With a power ratio of 4 when the Zl ratio is 9, ,it can be shown quite simply that the Ip,„ ratio is 2/3 and the Epn ratio is 6. I'hus, (Class-C min) Ip„ = Ibm/3 1—312 (31) and (Class-C max) = 3 Eco 1 — 312 (32) V2 P„„/4 _ Vp;7~ 2a)C.V 2wC a/ 2 R„ 1—312 (34) where Pcm is the true drive-level rating. Since (max) Ec will also be equal to Ece/2, approxi- mately, then (no longer continuing to treat as a magnitude only) (max) = - VT^ / V"2T57„ 1—312 (35) At the same time, Pzl ^ ^ must not ex- ceed Po„/4. Consequently, The plate dissipation in the tube should be of little concern unless subminiature tubes are used. In any event the plate dissipation is a maximum when Re is a maximum, so no thought need be taken for other than class-A operation. Approxi- mately, (Class-A) plate power = Ebib = EbIbm/2 1—312 (33) Finally, the foregoing equations suggest that a Pierce oscillator employing a sharp-cutoff tube be designed for class-A operation on the assumption that Re will be a maximum and that the maximum permissible drive level is one-fourth its actual rat- ing. Under these assumptions, equation 1-293(2) should be changed to (max) Ee = (max) Ib„ = - , = 2<cC. (max) I Eee I 1—312 (36) Equations (35) and (36) define the operating characteristics to be sought if a sharp-cutoff tube is to be used under conditions of maximum output for maximum stability. Remember, that equation (34) actually is an expression of the limitation on I„ the crystal current, and therefore upon Ep and E,. As far as the self-excitation voltage of a truly sharp-cutoff tube is concerned, it will be difficult to keep this voltage from building up until it reaches into the bend near the cutoff point. For this reason, the first consideration is that Ip™ is not exceeded. As a safety measure, Ii,m should not be greater than the value given by equation (36), even if the actual E™ is less than (max) E,„. The conclusions reached in the foregoing discussion are summarized in the following table. PIERCE-CIRCUIT OPERATING LIMITATIONS DUE TO CRYSTAL SPECIFICATIONS OF LOAD CAPACITANCE, C,, MAXIMUM PERMISSIBLE EFFECTIVE RESISTANCE, R,^.., AND DRIVE LEVEL, P.„ Plate Dissipation (max) = Ezlbm/Z E,o = — Pcm/ “Ox y/ i R,ni Ibm 2 P<,m/| Eco | Conditions are those for sharp-cutoff tube, negligible load, gridleak bias with large R^, C^C, = 1, Ep >> Ep, R,, > > (max) PI/4 = (max) Zl, Class-A operation when R^ is maximum, and maximum permissible output. R. = Rem R«n/2 R«o/9 Ep„, Ep„, 1 Ec 1 = 1 Ec./2 1 |Ec.| 3|Ec.| Pc (= Pzp) P.a./4 Pco./2 P™ Ipm Ibn./2 Iba./2 Ilini/3 Zb (min) PI/4 (min) PI/2 9 (min) PI/4 gm — 4/(min) PI 2/ (min) PI 4/9 (min) PI 6 = 0 IT ISw - 9 ~ Operation = Class A Class 6 Class C WADC TR 56-156 159 Sadion I Crystal Oscillator* Pierce-Oscillator Design Considerations When Tube Cutoff Has Below-Average Sharpness 1-313. Unless a vacuum tube has plate character- istics resemblinsr those of subminiature tubes when normal plate voltagres are used, or unless by reducing the filament voltage such characteristics can be achieved, a Pierce oscillator tube must be operated at a plate voltage of from one-half to one-fourth normal. In so doing, it is very probable that the lower bend of the EJo curve will become rather extended compared with the straight por- tion to the left of zero grid volts. In this event, the tube exhibits the characteristics of a remote-cut- off tube, except that the cutoff voltage is one-fifth or less that of a normal remote-cutoff tube operat- ing at an equivalent reduced plate voltage. Where the cutoff is not sharp, it is quite easy for equilib- rium to be reached with peak-to-peak excitation voltages much smaller in magnitude than Ecm and considerably greater ranges in R, of the crystal unit can exist before cutoff is reached. Thus, in the more usual case, the assumptions used in para- graph 1-312 cannot be made unless greater care is taken in the oscillator design to ensure a peak-to- peak excitation voltage equal to |E„| when R, is a maximum — an operating point much more difficult to locate and critical to maintain when a large steady decrease in the effective gm occurs well be- fore the cutoff point is reached, and which may require very low plate voltages if the maximum-R, crystal unit is not to be overdriven. As a concrete example, suppose that the oscillator is to employ a 10-mc crystal unit of the CR-18/U type. At this frequency, P<.m = 5 mw, R«„ = 25 ohms, and Cl = 32 /»/if. On substitution in equation 1 — 312 (35), we obtain a (max) of approximately — 5V. By equation 1 — 312(36), this value of Ec® is to be obtained in a tube where the zero-bias, with- out-signal plate current is Ibm = 2 ma. Such char- acteristics are not easily obtained with conven- tional-sized vacuum tubes. It may be necessary to operate at the given value of Ibn» or slightly greater, and a cutoff voltage that is of a smaller magnitude than that indicated for (max) E„ in equation 1 — 312(35) , in which case all crystal units used will drive the tube beyond cutoff. An alterna- tive approach is to operate at a larger than maxi- mum E„, but, if this be done, a safety factor should be allowed by assuming that Ip is to be the same for all values of R®. Although this will not be strictly true, the assumption is a close approxima- tion if the change in plate current between the values of E® = Eco/2 and E®® is very small com- pared with the change in plate current between E® = 0 and E® = E,.®/2. It can be seen that insofar as the effective Ip can be assumed to remain con- stant, Ep, and hence E*, I,, E®, and the crystal driving power, VR» =« Ip’Zl, increase directly with Zl, or inversely with R®. The problem is to find the maximum permissible E®, which, although apply- ing to extended-cutoff operation only when R® is a maximum, will not lead to a replacement crystal being over-driven if its resistance is less than the maximum. Again we assume a minimum R® equal to R®n,/9. In a manner similar to the derivation of equation 1 — 293(2), we can say (max) E® (with (min) R®) = (max) E® (for (min) R®] = 2 Re,n V2P®.® 2wCx\/ (min) R® 1—313 (1) Now, if equation (1) gives the bias voltage when a crystal unit of minimum R, is connected, assum- ing that Ip is constant, the bias that exists when a crystal unit of maximum R, (= 9 (min) R®) is substituted will be one-ninth the value above. Thus, (extended cutoff max) E® [for R® = R,m] = 1—313 (2) 3a;Cx %/ 2 Rem If a gridleak Pierce oscillator is not to have a loaded plate circuit, nor an adjustable nor con- trolled output voltage, nor a sharp cutoff, equa- tion (2) gives the maximum bias that can be safely assumed when R® is a maximum. The output volt- age agreeing with equation (2) is two-thirds that given in paragraph 1-312 for a sharp-cutoff tube. If Rg is not large enough for the average E® to ap- proximate the peak excitation voltage, a maximum bias less than that given by equation (2) must be assumed. With large values of R,, |E®| of equation (2) is the peak of the maximum excitation voltage when R® is a maximum, and |E®| of equation (1) is the approximate peak when R® is a minimum. If a Cg/Cp ratio other than 1 is used, equation (2) can be expressed more exactly (extended cutoff max) E® = WADC TR 56-156 160 — ! S«C, VTJ;; 1^13 (3) It should be understood that although equations (2) and (3) are derived from equation (1), it is wiser to select the vacuum tube and plate voltage upon the assumption that the resistance of the crystal unit is a maximum rather than a minimum. Since the effective amplification factor of the tube cannot be expected to be constant for all values of R„ equations (1) and (2) will not both hold for the same circuit. If (1) is correct, (2) will indicate a value too low; if (2) is correct, (1) will indicate a value too high. Equation (2) therefore permits a safety factor in the event of an exceptionally low value of Also, if the oscillator performs prop- erly with R, a maximum, it will almost certainly operate when R. is a minimum. The reverse is not necessarily true. 1-314. Equation 1 — 313(1) is equivalent to a bias and output of the same magnitude as that obtained in paragraph 1-312 for sharp-cutoff conditions and minimum R« ; but the bias and output of equation 1 — 312(2) for R, = Rm, when E„ is assumed to be significantly greater than 2Ec, are only two- thirds their equivalent sharp-cutoff values. In the case of the 10-mc CR-18/U crystal unit dis- cussed in paragraph 1-313, the (practical max) Ec, as given by equation 1 — 313 (2) is — « —1.7 V. For the smallest values of R., the bias will ap- proach —15 V. Assuming that gm ^ (according to equation 1 — 312(2), when C,/Cp = 1) and that Zl (min) PI 4 10 ‘® 4 X 6.28* X 32* _ 1 ^ " 4a)* C/ R.„ — = 2600 ohms X 25 then, gni 10V2500 = 400 ^unhos when R, is a maximum. This is a very small transconductance to be obtained with a bias of approximately —1.7 volts, and usually cannot be obtained at all with normal operating voltages except in the case of the small battery-operated tubes. The 1.7-volt nuiximum bias represents a peak-to-peak excitation maximum of 3.4 volts. With an average g„ of 400 )>mhos, the limiting value of Ibm (<«> 2Ipn 2g,aE,m) becomes 1.4 ma, approximately. Only if age is used to provide a much larger bias than can be obtained with a peak-to-peak excitation of 3.4 volts will it be possible to have such a small zero-signal plate current without operating conventional tubes at greatly reduced voltages. Generally, it is easier to operate with a small E,, and a larger Im and not WAOC TR 56-156 161 SecHon I Crystal Osdliotort attempt class-A operation. A large percentage of the crystal oscillators now in use drive the crystal units at a considerably higher level than is ad- vantageous from the point of view of stability and long crystal life. Much of the care otherwise taken in the circuit design can be wasted if the first consideration is power output rather than fre- quency control. Where a vacuum-tube manual recommends a particular voltage of power ampli- fier for use as a class-C oscillator tube, the typical operating characteristics listed are rarely appro- priate for military-standard crystal units, but apply more usually to LC circuits. The plate volt- ages must be considerably lower than the typical values indicated, in order to reach the small trans- conductances that must exist at equilibrium with- out overdriving the crystal unit. 1-315. Assume that a crystal unit is connected in a Pierce circuit using a conventional triode oper- ating at its normal plate voltage, and that the C,/Cp ratio is near unity. The equilibrium values of gm and Rp cannot be reached until the amplitude is great enough for the tube to be operating class C, and the crystal unit will almost certainly be over- driven. There are four ways in which the circuit can be adjusted to prevent this overdrive: (a) the plate voltage can be reduced, (b) the filament volt- age can be reduced, (c) the C,/CV ratio can be in- creased, or (d) the load losses can be increased. Of these methods, the first, reducing the plate voltage, seems to be the best from the point of view of frequency stability, although a reduction of the fila- ment voltage may be worth consideration. Very possibly, if the filament voltage is decreased suffi- ciently to lower the zero-bias transconductance to as much as one-fifth its normal value, the oper- ation of the circuit will become unduly sensitive to slight fluctuations in the filament power supply. The only data available at this writing is that re- ported by Messrs. Roberts, Novak, and Goldsmith of the Armour Research Foundation of Illinois Institute of Technology. Experimenting with a 6C4 tube and a 7-rnc Miller circuit, it was found that a 30-percent decrease in filament voltage, which is equivalent to decreasing the filament power by approximately one-half or more, depend- ing upon the temperature coefficient of the filament resistonce, caused only a 2.5-cycle rise in frequency. (In a Pierce circuit the frequency would have de- creased.) This effect on the frequency is very slight, but the exact decrease in the r-f plate cur- rent is not known. Nevertheless, the evidence is sufficient to suggest that if the tube character- istics are made suitable for a crystal circuit by reducing the filament voltage, any instability S«ction I Crystal Oscillators caused by further fluctuations in the filament volt- age would appear primarily as variations in the output voltage, rather than as variations in the fre- quency. In view of the fact that a reduction in filament current permits a greater saving in power than does a reduction in plate voltage (and length- ens the tube life), this approach to the problem may well be worth experimentation. The conven- tional approach, however, is to operate with a low plate voltage. If a C*/Cp ratio on the order of unity is to be used, the average tube will require plate or screen voltages of 40 to 50 volts, or less. The lower the voltage, the nearer class-A operation can be approached at equilibrium. A fair approximation of the operating conditions to be expected can be made from an inspection of a family of plate-char- acteristic curves. With Cg/Cp = 1, the peak-to-peak variations in plate voltage are the same as those of the excitation voltage, so for all practical pur- poses the plate voltage can be assumed to be con- stant. Thus, the load line can be assumed to be vertical, and the maximum and minimum ampli- tudes of Ip for a given plate voltage become the values, respectively, for grid voltages of 0 and 2Egm, where Eg^ is the peak excitation voltage. For the 10-mc crystal unit taken as an example above, it was found that the peak-to-peak I, for a maxi- mum Re was 2 Igre Eel = 1.4 ma. The correct plate voltage for a given tube is thus the value of Ep at which a change of grid voltage from 0 to —3.4 volts causes the plate current to decrease by 1.4 ma. This type of operation — class A to class AB — is generally more feasible when age is used, if it is desired to apply for all values of R,. PIERCE-OSCILLATOR DESIGN CONSIDERA- TIONS FOR Cg/Cp RATIOS OTHER THAN ONE 1-316. When the Gg/Cp ratio is not approximately equal to one but the total load capacitance meets the crystal specifications, gm is increased, and gen- erally it will be easier to obtain desirable vacuum- tube characteristics at more convenient plate volt- ages. The first step, as before, is to theoretically limit the peak of the crystal current to * / ^ , . \ Rgm When Re is a maximum. The peak excitation volt- 1 rsp age, Ega, equals “TT^/ “ 5 ^ under these conditions. Ep,„ equals equals C./CpZi,: I„m equals gmEgp,. ^ith these values taken as a start we can retrace the steps taken in paragraphs 1-312 and 1-313, and determine the values of Ipm and Ec® that do not permit the crystal to be overdriven for any value of R, between R«a and R,o,/9. FINAL WORD ON CORRECT LOAD CAPACITANCE IN THE PIERCE CIRCUIT 1-317. A prime purpose of the military specifica- tions regarding the load capacitance, effective re- sistance, drive level, and frequency tolerance of the different types of crystal units is to guarantee the replacement of a defective crystal unit in the field without special testing or other complications, and with the same ease that a defective vacuum tube can be replaced with a new tube of the same type. However, a crystal unit is more critical in its per- formance than a vacuum tube. As a result there can be no replacement guarantee unless the new crystal unit is inserted in a circuit where it will be operated under approximately the same load and drive conditions at which it has been tested. An inspection of the various types of oscillator circuits now in use, such as those illustrated in figures 1-135 to 1-138, most of which have been designed around the older types of crystal units, reveals a much greater versatility in operating conditions than is now desired in the design of new equipment. One of the requirements that is no longer within the jurisdiction of the design engi- neer is the effective load cajiacitance into which the crystal unit is to work. This means, that for a given nominal frequency and type of crystal unit, the crystal unit must exhibit a given inductive reactance, X*, equal numerically to l/<oC„ where C, is the rated load capacitance. Furthermore, it means that for each particular crystal unit there is but one frequency at which it is supposed to operate. This does not mean that all crystal units of the same type and nominal frequency have a single common operating frequency, rather that each has its own individual frequency, which, how- ever, will not differ from the nominal frequency by more than the permitted tolerance. It is the effective operating reactance that the crystal units must have in common. Approximately, X, = — 4irLAf — (Equation (1), figure 1 — 98) , , 4irLm Xco Now, Af = f, — f„ where fp is the operating par- allel-resonant frequency and f, is the series-reso- nant frequency of the motional arm. Assume that a 10-mc parallel-resonant crystal unit has a fre- quency tolerance of ±0.02 per cent. This is equiv- alent to an absolute frequency tolerance of ±2000 cps. Two crystal units at opposite extremes could be within specifications even though their oper- ating frequencies, fpi and fp„ were 4000 cps apart. WADC TR 56-156 162 If the rated load capacitance were 32 and the crystals were A elements, Af, itself, for each crys- tal would be on the order of 2000 cps. If the crystals were 6 elements of the same shunt ca- pacitance, Co, Af for each crystal would be only in the neighborhood of 800 cps, because of the B ele- ment’s larger series-arm inductance, L. It becomes obvious that there can be no expectation of “pull- ing” the frequencies together by making slight adjustments in the load capacitance. The lower- frequency crystal could not be raised to zero beat with a frequency 4000 cps higher without reduc- ing C. several-fold. The higher-frequency crystal could not even be “pulled” to the nominal frequency and oscillations still be maintained. For this reason, the design engineer should generally not attempt to provide an operator with frequency adjustments for the crystal oscillator. The only adjustments needed are those which can be factory preset, in order to compensate for slight differences in stray capacitance. If the frequencies to be gen- erated must be in close agreement with some standard, or with the frequency of some control- ling station, the task is to provide oven-controlled crystal units of smaller tolerance. Only when the desired operating tolerance is less than any pro- vided by crystal-unit specifications alone, is it necessary to provide the operator with a frequency adjustment knob. Even then, the adjustment need not provide a tuning range greater than the specified crystal tolerance. Since the smaller toler- ances are only 1/10 to 1/20 of the 0.02 per cent in the example above, the total variation in load ca- pacitance may not need to be greater than ±10 per cent of the specified capacitance. 1-318. It may be desirable to provide an operator with the means of controlling the output voltage Figure J-134. Ganged eapaeitances to enable adjuai- menf at C,/C, ratio at Pierce circuit wHhaut changing fatal load capacitance SocNon I Crystal Oscillators of a Pierce oscillator by varying the Cg/Cp ratio. In this case, care must be taken to ensure that the total load capacitance remains the same. If Cg and Cp are to be adjusted separately, some type of matching scales should be provided with the two tuning knobs, so that the correct load capacitance can always be had when, say, the two scales give the same reading. It is more desirable to have available ganged capacitors similar to those shown in figure 1-134 for each of the Military Standard capacitance ratings. The capacitors C, and Cj in series are to be designed to always ensure a correct load capacitance when each is shunted by conven- ient predetermined fixed capacitance. The small variable capacitances C, and C« are adjusted until the sum of their values and the circuit stray capac- itances, C.I and Cn, provide the correct fixed shunts for the ganged elements. MODIFICATIONS OF PIERCE CIRCUIT 1-319. Figures 1-135, 1-136, 1-137, and 1-138 and their accompanying circuit-data charts reveal a great flexibility in the design of a Pierce oscillator. It would be very convenient to be able to put our finger on a single circuit and say that the design of this circuit is superior to all others. Unfor- tunately, this is not possible. One would first have to define what is meant by “superior design.” The definition, at best would be a complex function of several physical and psychological variables. The very existence of a wide variety of circuit modi- fications suggests that no one circuit is superior to all others for all given tasks, although much of the variety can be attributed to the desire of the design engineer to create his own circuit and also to avoid the risk of possibly infringing upon the patent, rights of another. Our space does not per- mit a detailed discussion of each of the circuits shown. Only a few of the highlights are to be mentioned. In general, most of the oscillators illus- trated employ older-type crystal units ; most of the circuits use B-f voltages on the order of 200 volts, and would overdrive the smaller-sized crystals cur- rently recommended; and the load capacitances and the Cg/Cp ratios vary widely from one circuit to another. Those circuits that employ currently recommended Military Standard crystal units (crystal units having nomenclature type numbers, CR-15/U and higher) ' are designed to operate so that the crystal unit faces its rated parallel-mode load capacitance. In these circuits the plate sup- ply is normally 100 to 120 volts, and the crystal unit of average resistance is operated at 4 to 5 milliwatts. WADC TR 56-156 163 Circuit Data (or Figure 1-135. F in kc. R in kilohma. C in iiitf. L in nh. WAOC TR 56-156 164 165 Saclien I Crystal Oscillalort FIgun 1-135. Ceirfinwod Fig. Equipment Purpose F, F, F, CR R, R« R. (L) Receiver-Trans- mitter RT-173/ ARC-33 “Side-step” injector oscil- lator 7662 . 5 104,840- 192,280 F.-l-Pj CR-18/U 47 68 68 (M) Receiver-Trans- mitter RT-173/ ARC-33 Main channel local oscillator 12,517.8 15,325 im BfH 11 33 Lear Radio Set Model T-30AB- RCBBI^2 Local oscillator 655- 7155 455 IF. '68 0.68 5 4755- 3845 l■EOl FT-243 0.33 47 150 (P) Radio Set AN/ VRC-3 4300 (1st I.F.) ■REsI mi 0 33 47 150 (Q) Local oscilla- tor 1175- 8175 1700- 8700 Fe-F, (525 I.F.) 60 0.4 250 Circuit Data for Figure 1-135. F in kc. R in kilohms. C in pjtf. L in jih- WADC TR 56-156 166 WADC TR 56-156 167 Circuit D*ta for Figure 1-lSS. F in kc. R in kiiohms. C in L in ^h. WADC TR 56-156 16S S*cHon I Cryttal 0»cillat«r« R. R. C. C. C, wm ■1 a L, mm V. v. 24 100 3,000 m 500 1/2 12AT7 2000 B 27 470 iimii ■■ 27 n 2000 ■UiMI 20 830 3000 6 500 ■HRH m m 1-8 500 ■■ 155 15 MOPM ■UPS BQS2BI 22 Hi 1500 820 RFC m ■ ■ ■ 220 1500 1500 2 1500 4t 'A 5670 ■ WADC TR 56-156 169 Section I Crystal Oscillators TO FRCO MULTIPLIER SC4 >4 I P4 TO FREO D(VI0ER[(A)] TO FREO MULTIPLIER [(8)] FI 462.4SKC (C) B-f figure 1-136. ModiHcaiions of Pierce oscillator using screen-grid tubes Equipment Purpose F, Radio Set AN/ FRC-10 (WECo Transmitter D-156000) L-F osc for single-side- band opera- tion 625 Same as (A) H-F osc for single-side- band opera- tion 940- 5000 Switching Unit SA-107 ( )/MRC-4 BFO for two diversity re- ceivers 462.45 Radio Receiver R-270/FRR BFO 462.45 Radio Receiver BC-659-( ) Local osc of receiver 5675- 8650 R4 Rs 20 0 500 1000 1000 270 Circuit Data for Figure 1-136. F in kc. R in kilohms. C in mmI. L in fih. WADC TR 56-156 CT ( 0 ) FI (6‘9MC) CSCR ftgvr* 1-136. ConHnutd c , Cj Ca c , c . C, Cr 100 10,000 10,000 10,000 1000 ■ 50 Vari- able 10,000 10,000 10,000 1000 ■ 0 100 10- 100 100,000 75 20,000 39 10,000 100 100 95 100 5 1000 10,000 55 10,000 25 WADC TR 56-156 171 S«eli«ii I Crystal Oscillators Fig. Equipment Purpose n B CR R, R; R. o R. R. Rr (F) Radio Trans- mitter BC- 329-N M.O. 200- 400 ■ FT-249 100 0.22 15 10 66 5 0.022 (G) Lear Radio Set T-30AB- RCBBL-2 M.O. 2900- 6500 470 0.1 100 100 ■ 18 ■ (H) Modulator- Transmitter T-233/URW-3 M.O. (or remote-con- trol trans- mitter 3966 6- 5666 6 2Fi CR-IA/ AR 47 0.33 6 8 30 27 0 0051 ■ (I) Modulator- Transmitter BC-1158 M.O. 3966.6- 5666 6 2Fi CR-IA/ AR 47 0.33 6 8 30 27 0 0051 ■ (J) Radio Trans- mitter 12GLX-2 M.O. 260- 1750 F, Holder FT-249 100 0.1 50 385 B ■ ■ Circuit Data for Figure 1-136. F in kc. R in kilohma. C in mmI- L in uh. WAOC TR 56-156 172 WADC TR M-156 Section I Crycfal OtdHotore r3-*n OR «ri c . C , n L . V, ■ 1 , 10,000 10,000 10,000 18,000 18,000 6V6 GT/G 807 ■ 6U 6L6 45 470 470 1/2 815 1/2 815 45 470 470 1/2 815 1/2 815 100,000 100,000 Per F, 807 807 173 Section I Crystal Oscillators Figure I -137. E/ectron-coup/ed Fierce otclllator modMeations. All circuit* except circuit (A) provide frequency multiplication Fig. Equipment Purpose F, Fs F, CR Ri mm Rs R< R. Receiver-Transmit- ter RT-i78/ARC-27 Spectrum osc- illator 10,000 D CR-27/U 100 56 12 100 0 (B) Radio Receiver R-252/ARN-14 2nd monitor osc of receiver vfo 11,275- 11,725 M ■jm 8.2 68 ■ ■ (C) Radio Trans- mitter BC-400- CB,C,D,E) M.O. for mar- ker beacon 4166.67 2Fi FT-164 200 35 5 20 0 (D) Radio Trans- mitter T-67/ ARC-33 M.O. 5555- 8666 2F, 6Fi CR-IA/AR 5.1 H 47 47 0.083 (E) Radio Set AN/ARC-IA Main-cbannel heterodyne freq osc 2Fi 18F, CR-IA/AR CR-18/AR 100 8.2 ■ 100 (F) Frequency Meter TS-186/UP Crystal calibrator 5000 1 1 Navy CG- 40210; GE *G31 Ther- mocell. 6L6 tube envelope; heater; ± 0.002 per cent, —20 to 75*C 1 100 100 10 100 (G) Frequency Meter TS-186(B/C)/UP Crystal calibrator 5000 2Fi ■ CB-18/U (oven, 60*0 1.5 100 10 H Circuit Data for Figure 1-137. F in kc. R in kilohms. C in ttitl. L in <4h. WADC TR 56-156 174 WADC TR 56-156 175 Section I Crystal Oscillators Fig. Equipment Purpoee F, F, F, CR R. R. R. m ■a (H) Aircraft Radio Corp. Radio Transmitter ARC Types T-13 and T-llA M.O. 6000- 9000 3Fi 1 CAATC 11081 100 0.56 56 0.18 100 (I) Radio Receiver R-270/FRR Local oscUla* tor 1565= 8511.6 Fi, 2F„ or 3Fi ■ 47 27 ■ ■ ■ H Radio Trans- mitter BC-655- a,-am M.O. 5560- 8660 1 1 DC-ii- ( ). DC- 16, DC- 26. or CR-1( )/ AR 50 50 0.05 50 1 (K) Radio Transmitter BC-401-B M.O. jj^ll 100 0 51 Circuit Data for Figure 1-137. F in kc. R in kilohma. C in nitt- L in «•)>. WADC TR 56-156 176 vt Hgun I- 137. CanMnuMl R. Rt R. C, MM B 1 1 1 S5 120 120 40 (T-13) 50 (T-11) ■ ■ 100 9-140 10,000 10 1 ■ ■ 10 50 6800 6 250 6000 6000 6000 6 WADC TR 56-156 Svdion I Crystal Otcillatorf F2 F2 VZ ■ B a B B B mm 50 100 750 ■ 16,900 6AQ5 6AQ5 ■ ■ ■ m 1- 5.5 ■ ■ 2500 (60fl) 9-1/2 turns H H 000 6000 1000 350 2500 m HI 177 Section I Crystal Oscillators Circuit Data (or Figure 1-137. F in ke. R in kilohms. C in mit. L in jih. WADC TR 56-156 '7B FIgwi* 1-137. CouHnomd R. 1 Ri wm a m ■ ■ 25 10,000 3000 6-42 (eaci sect- ion) 100 ■ 25 100 3000 6-42 (eacl sect- ion) ■ 20 220 2000 10 0 1 27 28 33 565~ 100 10,00 ■ ■ 1000 10 15 WADC TR 56-156 Section I Cryttol Otdilators R7 (P) 1 D c. C: ID D D D V. li 1000 ■ 1700 (40fi) ■ 6SK7 9003 1 1000 ■ 9.3 ■ 9003 8000 90 103 5 500 H 0 3 10,000 1.5-7 5639 Balanced modulator 220 10,000 60,000 500 ■ mmn 179 Section J Crystol Oscillators R3 (0) (R) (S) FI I3.02S KC CR‘I6/U FIgvn f-f37. ConflniMd Fig. Equipment Purpoae F. D (Q) Radio Set AN/ARC-34(XA-1) 2nd monitor osc of transmitter M.O. 3230-3900 (5 crystals) F, (R) Radio Set AN/ARC-34(XA-1) 3rd monitor osc of transmitter M.O. 3650-3775 (4 crystals) Pi (S) Radio Set AN/ARC-84(XA-1) 4th monitor osc of transmitter M.O. 5130-5165 (6 crystals) Pi (T) 13,025 m (U) Radio Receiver R-322/ARN-18 Local osc 11,490- 11,700 (20 crystals) (V) wm Coarse fre<^ osc for mixing with output of fine freq oac 4800-9600 (28 crystals) 3Fi Circuit Data for Figure 1>137. F in kc. R in kilohms. C in fifd, L in WAOC TR 56-156 180 ISO V •H30 V V2 SUPPRESSOR GRID IN (Q) ANO (R) CONTROL GRID IN (S) TO (T) D CR R. Ri R. R. R. ■ CR-27/U 560 56 B 1.8 0.68 ■ 560 56 m 1.8 () ■ gnu 560 56 m 1.8 0 560 0.56 ■H ■ 0.01 ■ B ■ ■ CR-18/tJ' 100 0.15 100 ■ ■ Mh. - ' ' ! Section I •» v"’-,-,'.; Cryitol Ascillcrtert •I n S4.#T0-3S,I00KC R. Rt Ri a C, D B D MM D D L, a V, V, 1 0.47 10 47 1000 10 33 10,000 470 ■ 22 5000 ■ 5840 5636 0 ■ 10 47 1000 10 33 10,000 12 0 5000 5840 6636 0 ■! H ■n 10,000 470 10 M 5840 6840 m 22 ■■ 10,000 10,000 1000 130 500 5840 ■ ■ 6 ■ ■ ■ m ■ ■ ■ 6 ■ ■ m ■ 1500 ■ 500 6AK6W 1 WAOC TR 56-15« 181 Section I Crystal Oscillators F2 C3 n IB) Figure 1-138. Flectron-coupled Pierce oscillator modifications for heterodyne circuits Fig. Equipment Purpose F, Fa Fa CR Ri R2 (A) Radio Receiver R-262/ARN-14 1st monitor osc of receiver vfo. 14,000- 20,500 From iscH lation amplifier of vfo. n CR-33/U 10.4 10 (B) Signal Generator TS-413/U Crystal harmonic heterodyne mixer 1000 Variable rf ISIS igim DC-9-AJ 1000 35 (C) Radio Receiver R-146A/AKW-35 2nd heterodyne osc 5456 5000 FT-243 47 47 (D) R-F Signal Generator Set AN/URM-25C Crystal calibrator harmonic generator and mixer 1000 10-50,000 (output from variable osc) CR-18/U 1 270 (E) — Radio Receiver R-277 (XA-A)/ Al’N-70 Local osc 400 480 2850 2950 3000 3050 F.-Fs 300 HOO (IF) CR-26/U (LF,) CR-18/U (HF.) 22 0.068 Circuit Data for Figure 1-138. F in kc. R in kilohms. C in fi/if. L in uh- WAOC TR 56-156 182 Section I Crystal Oscillators FS Rs R4 Rs D C^ Ca C, Cs c. L, U La V, 47 22 6.8 68 27 470 47 1800 ■ 1 5725/ 6AS6 100 10 330 5600 10,000 1 6SA7 47 47 25 5-30 55 50,000 12SA7 30 100 510 1000 6-25 270 10,000 10,000 2500 500 6BE6 22 47 470 100 47,000 5750 WADC TR 56-156 183 section I Crystal Oscillators Preventing B+ Voltage from Existing Across Crystal Unit in Pierce Circvit 1-320. There are two commonly used methods for preventing the application of the B+ voltage across the crystal unit. One method is to connect a blocking capacitor in series with the crystal unit. The other method is to connect the crystal unit directly to ground, and the plate to ground through an r-f bypass capacitor. With this arrangement, the cathode must be operated above r-f ground. Except for the advantages of electron-coupling to the load to be had in the tri-tet circuit, which uses screen-grid tubes, the r-f-grounded-plate modifies^ tion is not to be preferred since the plate-to-grid capacitance directly shunting the crystal unit is greatly increased by the addition of the grid-to- ground capacitance. When a capacitor is used to block the B+ voltage from the crystal, a number of modifications are possible, three of which are shown in figures 1-135 (F) , (G) , and (K) . In each figure, the blocking capacitor is the one labeled Cj, and has a reactance negligible with respect to that of the crystal unit. The more usual circuit arrange- ment is that shown in figure 1-135 (K). In figure 1-135 (G), Cj plays a dual function in blocking the B+ voltage from both the crystal and the grid circuit of the succeeding stage. The circuit in figure 1-135 (F) , although not effective in block- ing the B+ voltage from the crystal unit, does, of course, reduce the d-c potential across the crystal equally as well as the other arrangements. It is desirable to keep the d-c potential across the crys- tal unit low ; otherwise the crystal may be heavily strained in one direction, and the chance is in- creased that the elastic limit of the crystal will be approached on the alternation of the a-c voltage of the same polarity, or, at least, that the effect on the crystal will cause the performance char- acteristics to deviate from test specifications. Since the crystal unit, itself, is a capacitor of consider- ably greater dielectric thickness and smaller cross- sectional area than the usual blocking capacitor, it might seem questionable that the d-c voltage across the crystal unit could be expected to be significantly reduced. Certainly in the static state, an air-gap crystal unit should have at least as high a resistance to leakage currents as would the block- ing capacitor. However, in the dynamic state some degree of ionization will occur if the voltage is ex- cessive. The consequent leakage tends to charge the blocking capacitor to the full plate-to-grid d-c voltage. An unchecked B+ voltage not only can in- crease the likelihood of corona effects, but also can lead to continuous discharge should corona losses once begin, and to an increase in the effective re- sistance of the crystal unit, a reduction of the grid bias, and perhaps even to arcing and puncturing of the crystal. A blocking capacitor, if it does not remove the d-c potential completely from across the crystal unit, at least can ensure that the po- tential is not sufficient to cause ionization. A block- ing capacitor is generally more important in high-drive circuits employing air-gap crystal units. 1-321. Figure 1-135 (X) shows one example of a grounded-crystal, grounded-plate Pierce circuit. At first glance such an oscillator might very easily be mistaken for the Miller type. C* is a relatively large capacitance that causes the plate to be at r-f ground. Ci is effectively Cp, the plate-to-cathode capacitance, and Ci serves as the lumped part of Cp. By interchanging Ci and C, and making the ground connection of the crystal a plate connec- tion, essentially the same oscillator characteristics are obtained except that the B+ is across the crys- tal unit. Note that the circuit is designed for the maximum possible output voltage, in that the out- put is taken across the crystal unit directly, in- stead of across Ci, the effective plate capacitance, alone. The fact that the plate is at r-f ground does not remove the grid-to-plate capacitance from shunting the crystal unit, but adds to this the grid- to-ground capacitance. In the circuit of figure 1-135 (R), a small tuning capacitance, C,, is also shunted across the crystal unit. In addition to this there is the extra shunt capacitance contributed by the oven in which the crystal is mounted. In cir- cuit (R), the Ci/Cj ratio, which is approximately equal to the C*/Cp ratio, is on the order of %. The gridleak losses are increased somewhat, since the grid resistance is connected across the crystal in- stead of across the grid capacitance. All these factors tend to reduce the effective Qf of the feed- back circuit, so it would appear that with crystal units of greater than average resistance the cir- cuit operates with the tank considerably detuned from resonance. The fact is, however, that con- necting the grid resistance across the crystal serves to concentrate most of the grid losses in the plate-to-grid circuit and to eliminate them from the grid-to-cathode circuits, and this probably in- creases the effective feed-back Qf more than the increased losses diminish it. The circuit in figure 1-135 (X) is similar to that in figure 1-135 (R) except that the tuning capacitance has been elim- inated and the output is obtained directly across the crystal unit. It is claimed that this arrange- ment tends to smooth the output and to reduce the harmonics. What probably is meant is that for a given output voltage the harmonic content is less. This can readily be seen, for the voltage across WADC TR 56-156 184 the crystal unit is equal to the sum of the voltages across Ct and Ci. If the same output is to be taken across either capacitance alone, the excitation must be increased and the circuit will generally be operated beyond tube cutoff a greater fraction of the time, thereby increasing the hig^r-harmonic content. It should also be noted that the output arm in figure 1-135 (X), since it shunts the crys- tal, is effectively part of the feed-back circuit. Should the load increase or decrease, so also will the excitation. The circuit is thus a tri-tet modifi- cation where the output voltage is somewhat stabilized against changes in the load, but only at the sacrifice of frequency stability. This feature is not important in the particular fixed-load cir- cuit of figure 1-135 (X), since the load in this circuit appears to be reasonably constant. Electron^oupled Pierce Oscillator 1-322. The electron-coupled oscillator permits a re- markable freedom from coupling between the plate load circuit and the crystal circuit. Screen-grid tubes are required, with the screen grid serving as the plate of the oscillator circuit. When electron coupling is used in conjunction with Pierce oscil- lators, the tri-tet arrangement, where the plate load circuit is in series with the oscillator tank, is generally the most advantageous, and is used in all the circuits shown in figure 1-137 except in circuits (I) and (0). With the screen at r-f ground, the vacuum-tube plate circuit performs as a conventional pentode r-f amplifier, with the ex- citation of the control grid a function of both the screen and plate r-f currents. Variations in the plate impedances have much less effect upon the frequency than do similar variations in the oscil- lator tank impedances. For this reason, even if the oscillator is to operate over a wide range of crystal frequencies, a tunable coil and capacitor tank can be placed in the plate circuit to obtain a smoother sine-wave output without running the risk of greatly changing the load capacitance of the crys- tal circuit. In effect, the electron-coupled oscillator reduces by one the number of amplifier stages that are required, and hence is particularly applicable for small portable transmitters where the crystal circuit must perform as nearly as possible the func- tion of a power oscillator. The widest application of the electron-coupled Pierce circuit is for the pur- pose of frequency multiplication. In figure 1-137 (D), for example, the plate tank circuit is tuned to twice the crystal frequency. The L/C ratio of the plate tank should be as small as practicable, in order to increase the output selectivity and to en- sure a low-impedance bypass through the coil for the fundamental frequency and through the ca- Section I Crystal Osdllalors pacitor for all harmonics higher than the second. The larger the angle 6 during which the tube is cut off, the larger will be the percentage of the higher-harmonic generation in the output. In gen- eral, the lower the order of the harmonic, the greater is its energy content. For optimum output, the tube should not be heavily conducting during a positive alternation of the plate harmonic volt- age, Eh. During such intervals the plate tank would be losing energy to the circuit at an instantaneous rate of ibOb, where ib is the instantaneous d-c plate current and eb is the instantaneous harmonic volt- age across the plate tank. To meet the require- ments above, plate current should be allowed to flow only during the interval of approximately one alternation of a harmonic cycle. Since the tube is to be cut on and off at the fundamental frequency, the plate tank, after receiving a pulse of energy dur- ing one alternation of a harmonic cycle, must oscillate freely for the remaining part of the funda- mental period. If the frequency is being doubled, plate current should flow approximately one-fourth the time; if the frequency is being tripled, plate current should flow approximately one-sixth of the time. To generalize, if the frequency is to be multi- plied n times, optimum n’th harmonic output is approached if the oscillator is designed so that the tube conducts approximately l/2n of each funda- mental cycle. In paragraph 1-312, it was found that for a given peak value of ib, (Ibm) , the effective Ip was essentially constant for all bias voltages be- tween class-A and class-B operations, although a small maximum occured when the tube was cut off during three-fifths of the negative alternation. See equation 1-312 (20). If the plate load is con- stant, as is the case in the electron-coupled circuit, this point of maximum Ip coincides with the condi- tions of maximum output. If we assume that ap- proximately the same conditions hold in the case of frequency multiplication, maximum harmonic output is approached if the tube continuously con- ducts 7/10 of the period of one harmonic cycle, or 7/lOn of the period of the fundamental cycle. Since the maximum is not at all sharp, the optimum operating conditions are not critical and can be assumed to extend over a range which permits the tube to conduct from l/2n to 7/lOn of the time. That is, for optimum output, the oscillator section can be designed so that the crystal unit of average resistance allows the tube to be cut off during a fundamental-cycle angle within the range given by - 1 ) to n 5n 1—322 (1) WAOC Tt 56-156 1«5 S«ction I Cr/ctal OscillQtQrs The principal advantage of the tri-tet circuit is that the excitation voltage tends to increase with an increase in load. If the output is to be inductively coupled to the succeeding stage, the tri-tet ar- rangement tends to stabilize the output voltage when the coefficient of coupling is varied. This fea- ture originally found its greatest popularity among radio amateurs, although it was used principally in conjunction with Miller rather than Pierce oscil- lators. Figure 1-139 shows the basic tri-tet circuit as applied to Pierce and to Miller oscillators. It is the adaptability to frequency multiplication rather than to variable load conditions, however, that is of greatest importance when considering the tri- tet circuit for use in militarj equipments. The tri- tet frequency stability is low compared with that of the conventional pentode circuit because of the large stray capacitance (approximately 8 ^t^f ) that directly shunts the crystal unit. About 4 mti is the of the tube, and the remainder is the capac- itance of the grid leads to ground, which otherwise would be part of C,. Miscellaneous Pierce Circuit Modifications 1-S23. Circuits (A) and (B) in figure 1-135 are of interest because they indicate two stages in the development of a particular modification of the Pierce oscillator. Originally Ls, C„ C4, and R4 were not present. Since no grid capacitance is employed other than that of the tube, the C,/Cp ratio is very small. The tube has approximately 4 md capac- itance between plate and grid, and it may be as- sumed that the crystal oven adds a comparable amount directly across the crystal. In all proba- bility the phase-shifting Q, of the feed-back circuit is rather low, so that the tank can be expected to be more reactive than resistive. L, is resonant with Cl at the mid-point of the intended frequency range. If the feed-back circuit is operated with a high effective Q*, this value of L, should provide a zero phase shift in Ip. In this event, the tube would operate into a resistive load, and a theoretical in- dependence of the frequency with changes in Rp could be predicted. As it is, the low feed-back Q, lA) PIERCE TRI-TET ( B) miller tri-tet Figure 1-139. Basie tri-tet circuits where excHaflan voltage Is a function of both screen- and plate-circultr-f currents WADC TR 56-156 186 probably requires the crystal tank circuit to be de- tuned to such a point that at equilibrim it appears either as a reactance much greater in magnitude than that of the coil L,, or as a reactance approxi- mately equal to that of Ci, in which case the lagging component of the current through the crystal unit is negligible compared with the current through Cl. Under these latter conditions, the tube would operate into a low-impedance, series-resonant cir- cuit at the mid-point of the frequency range, where Xli + Xoi = 0 .- Nevertheless, it was found that oscillations could not be maintained dependably at the mid-point of the frequency range. It was for this reason that the changes were made in the models represented by circuit (B) , as indicated in the data chart for figure 1-135. The tank circuit L 3 C, is resonant at the mid-point of the frequency range. C, has been changed so that the reactance of the coil L, is lower than the reactance of the crystal tank at all frequencies. R 4 has been added to dampen the effect of L, at the high end of the frequency range. It may be that the dead spot at the mid-point of the frequency range in circuit (A) was due only to transient effects in the crystal units before oscillations could build up, or it may have been due to the fact that the feed-back Qi was insufficient to provide the necessary plate im- pedance to maintain equilibrium even if oscilla- tions were once started. 1-324. An interesting circuit is that shown in fig- ure 1-136 (D). The feed-back voltage is developed across C, by the r-f plate current. Cs, although of the same capacitance as C 4 , maintains the screen at r-f ground, since R, is very large. The large value of R, keeps the screen voltage, and hence the out put, at very low values, so that the crystal is only weakly driven. 1-325. The circuit shown in figure 1-136 (E) is Section I Crystal Oscillators intended to supply a fourth-harmonic excitation of the V» stage. For this purpose the CjLa tank is tuned to 4F,. A low Lj/C, ratio is provided, to en- sure that the fundamental is effectively bspassed. The capacitance C» is kept small so as to present a high reactance to the fundamental, else the fun- damental would be entirely bypassed around the crystal circuit. 1-326. The circuit shown in figure 1-136 (G) is something of a novelty in that a Pierce instead of a Miller oscillator is employed to directly excite the power amplifier of a small transmitter. The L,C 4 arm is a neutralizing circuit which prevents the amplitude-modulated output stage from varying the effective impedance of the oscillator load. Nor- mally, neutralizing networks are not necessary for crystal oscillators. Only when the oscillators drive power amplifiers directly is feed-back neutrali- zation advisable. Even then, if the power amplifier is not modulated and performs as a frequency multiplier, neutralization is not necessary. 1-327. The electron-coupled converter circuits shown in figime 1-138 embody more or less the same features previously discussed. The basic methods illustrated for obtaining a heterodyne out- put are more or less self-explanatory, and will not be elaborated upon here. The Miller Oseillafer 1-328. The Miller oscillator is the crystal equivalent of a Hartley oscillator in which no mutual induct- ance exists between the plate-to-cathode and grid- to-cathode inductances. ( See figure 1-140.) The Miller oscillator has an average frequency devia- tion of approximately 1.5 times that of the Pierce circuit. The plate circuit must appear inductive in order that the correct phase shift will be produced in Ep, the plate r-f voltage, to compensate for the t Zpg \ 2g Figun WAOC TR (A) MILLER OSCILLATOR 1-140. Diagram* Ulu$tratlng tha agulvalanca oatwaaa th» Millar tirtult and tha Hartley tircuil 56-15A 187 Section I Crystal Oscillators resistance in the feed-back arm, since this resist- ance prevents the necessary 180-degree phase rotation of the equivalent generator voltage of the ampliher from occurring entirely in the feed-back circuit. The effective load capacitance into which the crystal unit operates is, approximately. C. = c. 1—328 (1) where Z„g and Zp are both considered as unsigned magnitudes, and the various symbols correspond to those in figure 1-140. Since the load capacitance is a function of the frequency, a Miller oscillator cannot be operated at more than one frequency and still present the same load capacitance to each crys- tal unit except by providing for an adjustment of the circuit parameters. In spite of its greater fre- quency instability and lack of circuit simplicity as compared with the Pierce circuit, the Miller de- sign is the one most widely used in crystal oscil- lators. The reason for this popularity is the greater output that can be obtained for the same crystal drive level. In either the Pierce or the Miller basic circuit, the output cannot exceed the voltage across Zp„ the largest single impedance in the plate tank circuit. In the Pierce circuit, the maximum voltage is thus the maximum permissible across the crystal unit ; in the Miller circuit the maximum voltage is (k -|- 1) times the maximum permissible voltage across the crystal unit, where k is the gain of the stage, equal to Ep/E*. This gain, theoretically (not practically), can approach the mu of the tube as a limit when the load impedance, Zl, is large com- pared with Rp. Thus, the use of a Miller circuit permits a saving of one amplifier stage. 1-329. The feed-back capacitance of the Miller cir- cuit is, normally, simply the plate-to-grid inter- electrode capacitance of the tube. It cannot, of course, be less than this unless an inductive shunt is connected between the plate and grid. When a pentode is used, it is usually necessary to insert a small feed-back capacitance on the order of a few micromicrofarads. The waveform in the output is improved by the use of a tuned tank circuit having a low L/C ratio in place of Lp. The plate tank must be tuned to a frequency above the oscillator fre- quency, in order that the tank impedance will ap- pear inductive. Such an arrangement also ensures a large effective Lp of high Q. A variable capac- itance in the plate tank facilitates adjustments to obtain the correct load capacitance for the crystal unit. MILLER-OSCILLATOR DESIGN CONSIDERATIONS 1-330. If it is decided to employ a Miller oscillator as a frequency generator, the choice should be dictated by the need of a greater output than can be obtained with a Pierce oscillator. An exception to this rule might be made if a tri-tet circuit is contemplated, in which case, the large capacitance that will directly shunt the Pierce-connected crys- tal may well prevent the stability from being as high as that of the Miller tri-tet circuit. The Miller circuit is the more critical to design insofar as maintaining the correct load capacitance is con- cerned, but the basic approach to the problem is the same as that which was followed in analyzing the equilibrium state of the Pierce circuit. Both oscillators are represented by the same basic cir- cuit, shown in figure 1-119. We shall not repeat the steps involved in the derivation of the equilibrium equations in the particular case of the Miller oscil- lator. The basic equations given in the following paragraphs can be used as points of departure in the design of any Miller circuit. Also, by methods similar to those employed in the analysis of the Pierce circuit, the design limitations of a Miller oscillator in which the crystal unit is to be oper- ated within specifications can be predetermined, approximately. MILLER-OSCILLATOR EQUATION OF STATE 1-331. As in the case of the Pierce circuit, there are two equations that express the state of oscil- lation equilibrium in the Miller circuit. Originally derived by Koga, these two equations are the real and the imaginary parts of the general equation: MZpZg Rp Z, -p Zp(Zg Zpg) where 1-331 (1) Z, = Zg -|- 2^ -4- Zpg and Zg = Rpg -f- jXgj Zp = jXp; Zpg = jXpg. R,,, is the effective resistance of the grid circuit, accounting for both the crystal and gridleak losses. The losses in the plate circuit are assumed to be negligible. On solving equation (1), the real part can be expressed as ^ Xp [Z,^ (M + 1) + X, Xpg] Reg (Xp -t- Xpg) WADC TR 56-156 188 1—331 (2) Equation (2) defines the conditions that exist when the feedback power input equals the power dissipated in the grid circuit. The imaginary part of the equation (1) defines the frequency, or, more exactly, the impedance relations that must exist if the feedback is to be of proper phase. This is given as X, = 1-331 (3) Where is equal to X»/R,.g, and X, = X,, + X* -f X,«. Equation (1) is the same 'Ss equation 1-289 (1) except that the terms are rearranged and ^ is substituted for the product R,.g,„. Equations (2) and (3) correspond to equations 1-289 (2) and (3) , respectively. If it is assumed that Z, >=« X„ and that (Xp, -|- Xp,,,) != —X,,, equation (2) above can be simplified, thus; R>. Xp, [X, (m + 1 ) + Xp„l R..„ 1—331 (4) Remember that Xp. and X^ are positive, and that X,.,, is negative. When equation (3) is rearranged as follows Xp. + X, -F = - Xp. 1—331 (5) it can be seen that the effect of the tube Rp on the frequency is a function of the term LOAD CAPACITANCE OF CRYSTAL UNIT IN MILLER OSCILLATOR 1-332. The load capacitance into which the crystal unit operates in a Miller circuit has been derived by Koga to be C. = Cp. -t- Cp. -I- Cv 1—332 (1) where C. and C,,. are as represented in figure 1-140, and Section I Crystal Oscillators latter equation in the following form : T + ^ + —7— R r = 0 ■ " + 1—332 (3) It is next assumed that the term in parentheses 1 -I- Rp/Zp -f — = 1 -f Rp/Zp 1—332 (4) Such an assumption not only implies that the leed- back current is negligible compared with the r-f current through the plate coil, L,,, but that the feed-back impedance is so high relative to Rp that Ri,/(Zk -f- Zp,,) is negligible compared with 1. The former implication requires that Z,., the effective load impedance across the tube, be approximately equal to jX,,; the latter implication requires that X,,p,| — IXp,! >> Rp. If the effective phase-deter- mining Qf of the feed-back circuit is 10 or more, as is very likely to be the case when standard crystal units are operating at their rated local load ca- pacitance and Cp. is not excessive, then Ep must be very nearly in phase with — mE.. Such a condition cannot exist simultaneously with equation (4) un- less Rp, < < Xp — an operating state that would be very undesirable from the point of view of fre- quency stability. If R,, is to have a reasonable value at the rated load capacitance of the crystal unit, the impedance of the feed-back arm cannot be greatly different from that of the plate circuit. Equation (4) would be sufficiently accurate for very low values of Q. and very large transconduct- ances for the tube; however, it would seem that for crystal units that are to be operated well above series resonance, the approximation of equation (4) should not be made. In this case 1 -t- Rp/Zp + „ = 1 + Rp/Zl "g “T ^pg '1—332 (5 It appears that equation (2) gives a value to Cr that, for a Miller oscillator operated at the rated load capacitance of the crystal unit, is probably between three and four times too small for the average circuit. Equation (2) is derived from equation 1-331 (1), beginning by expressing the and Zi, approaches Xp.^/R,.. as R,, and Q. increase. Since Zt., as used above, represents an involved complex quantity, an exact expression of equation (2) will not be attempted here. As can be seen from equation 1-331 (5), if 1/Q. and X„/Rp are each on order of 1/10 or smaller, X. -f Xp, =» 1X,,.|. On the other hand, if Xp,/R,, is not small, the vari- able parameters of the vacuum tube and the varia- tions to be expected in the effective resistance from WADC TR 56-156 189 Section I Crystal Oscillators one crystal unit to another will have such a large influence upon the effective load capacitance that there can never be an assurance that a crystal chosen at random will be operated according to specifications. In other words, a Miller oscillator cannot be designed to provide approximately a specified load capacitance unless X* + X,, |X,,,|. Under these conditions C, = C„ -1 — = C* + Cp, + Cx„ 1—332 (6) where It must be understood that equations (6) and (7) assume that R,, is large compared with X,„ and that Xg is large compared with R. ^.. For this latter con- dition to hold, the grid-to-cathode capacitance, Cg, must be kept as small as possible. If the assump- tions above cannot be made, it is not feasible to expect a Miller oscillator to operate at approxi- mately the same load capacitance for all crystal units, nor can good frequency stability be ex- pected. A more comprehensive capacitance equa- tion for the Miller circuit — one that holds approxi- mately for all operating conditions — can be ex- pressed as C, = Cg -I- Cpg -h Cxp' 1-332 (8) where Cx,.' is given by equation (7), except that X|, is replaced by X,,', where Xp' = Xp + R.g/Qg - 1^2 (9) It will be seen that X,,' has been so chosen that equation 1-331 (5) can be expressed in the form where P„ is the power dissipated in the output circuit, and = EgVP* 1—333 (2) where P^ is the power dissipated in the grid cir- cuit. If the gridleak losses are negligible, P,, equals the crystal power and Rg,. equals the PI of the crystal unit. As can be seen from the equations in figure 1-141, either PI must be small or R, very large for this assumption to hold. With Rp large compared with Zj,, Ip is approximately equal to gmEp. We shall assume that X, = X^ -f Xp + X,* « 0. Under these conditions it can be shown quite simply that in the circuit of figure 1-141, letting k = Ep/Eg, k" R„ + Rp k R„ R„ 1—333 (3) For a given Rg,. and Ro, equation (3) has a mini- mum g,„ when k* = Ro/Rgg 1—333 (4) Since a maximum Rp coincides with a minimum gm, equation (4) also establishes the conditions for a maximum Rp. Now, Rg, is a function of R, of the crystal unit, so that a circuit design using equa- tion (4) should be based on a most probable value of Rg, (i.e., a most probable value of R,), which will usually correspond to a value of R, between one-third and one-fourth of the maximum R,. Equation (4) should not be interpreted to mean that if Ro/Rg, is adjusted to equal a fixed value of k*, the g„, of the tube will therefore be a minimum relative to its values for other R,/Rg, ratios. Such an interpretation would only hold true if the prod- uct R„Rg, were constant. Where equation (4) holds, it can be shown that the ratio of output power to crystal power is P„/Pg = 1 1—333 (5) Xg -h Xp' + Xpg = 0 1-332 (10) In the event that the plate circuit contains a ca- pacitance shunting the coil, equations (6) and (7) still hold except that Xp refers to the total parallel reactance in the LpCp branches. MAXIMUM Rp OF MILLER OSCILLATOR TUBE UNDER GIVEN LOAD CONDITIONS 1-333. Referring to figure 1-141, Rp and R*, are defined as follows ; Rp = EpVPp WADC TR 56-156 OPTIMUM VALUE OF k - EJE, FOR MILLER OSCILLATOR 1-334. Practical values of k, unless a pentode is used, are limited by the plate-to-grid and grid-to- cathode interelectrode capacitances of the tube and the specified load capacitance of the crystal unit. If X. = Xg -f- Xp -f Xpg « 0, then Xp = kXg 1—334 (1) and 1—334 (2) 1—333 (1) 190 - Xp, = (k + 1) Xg Section I Crystal Oscillators figure U141. Equivalent circuit of Miller oscillator. Up is assumed to be large compared with the total load impedance. Il„ is an equivalent resistance accounting for the output losses. R,„. is an equivalent resistance accounting for the crystal and grid losses; it is approximately equal to the resistance of the parallel circuit shown in (Bi. PI is the performance index of the crystal unit; K„' is the equivalent grid resistance; and R„ is the actual gridleak resistance By equation (2) . — (Xp* + X,) x;^ where 1—334 (3) X. = — l/«Cpg , Cx = rated load capacitance, and = grid-to- cathode capacitance. With triodes, values of k above 4 or 5 are difficult to obtain. If the oscillator is to be^ designed with no other feedback-circuit capacitances than those provided by the interelec- trode capacitances, 0,.* and C^, of the tube, it can be assumed that k is a fixed parameter equal to the value given by equation (3). The output arm must thus be designed to provide a reactance, Xp =« kXg, if the crystal unit is to operate into its rated load capacitance. 1-335. From the point of view of frequency sta- bility it is desired that the term (XpXpe/RpQ*) in equation 1 — 331 (5) be as small as possible rela- tive to (X, 4- Xp), or, equivalently, to IXpgj. In other words. WADC TR 56-156 should be a maximum. With equation 1 — 331 (4), it can be shown that Rp/k = Q,X,(m - 1) 1-335 (1) Equation (1) indicates that as the fraction of the loop reactance (kR,,,/Rp) dependent upon Rp be- comes smaller, the effective amplification factor of the tube becomes greater. Intuitively from equa- tion (1) it can be seen that with Q^X^ constant for a given vacuum tube and plate voltage, Ei„ R,, must increase as k is made larger, otherwise ^ could not decrease. Nevertheless, the larger that k be- comes the smaller the value of Rp/k. If k = 1, a frequency stability almost approaching that of the Pierce circuit can be achieved, but with twice the output voltage. Lower values of k would soon de- tune the oscillating tank to a point where the simplifying assumptions made regarding k would no longer hold. 1-336. Since the principal purpose of using a Miller instead of a Pierce circuit is to eliminate an am- plifier stage, and since Ep is limited to the maxi- mum voltage that can be placed across the crystal unit, the ratio Ep/E^ = k can be chosen to give a desired gain over that which would be o''tained with a Pierce oscillator operating at the same crys- tal drive level. Ej in the Miller circuit caa-be as- sumed to be twice the Eg of a Pierce circuit that 191 Saction I Cryttol Oscillator* has a k = 1. If an imaginary gain of 10 is desired, k for the Miller circuit should be equal to 5. For k to be 5, according to equation 1 — 334 (3) Cp* = (C, - C«)/6 If Cx —32 fifit and Cf = 8 C,* must be 4 ji/if, which is a value quite representative of the aver- age triode amplifier, or which could be obtained with a pentode by using a small external plate-to- grid capacitance. Equation 1 — 335 (1) can be re- written in the form l/Rp = g,n - Rc«/kX,= 1-336 (1) For k = 5, the value of l/R,,, and hence the per- centage effect of R|, on the loop reactance, will be a minimum the more nearly that g,„ can be made to approach in value R.^/kX^* = R,.„/5Xg'. If the X effective Q of the crystal unit, Q, =; is equal to 10 or more and if the gridleak losses are negligible, it can be shown that and that X, = 1-336 (3) Ag -f- A(;^^ Thus, Q, = 1_336 (4) Re Xc* and X,VRc« == XeVRe = PI 1—336 (5) where PI is the performance index of the crystal unit. If it is further assumed that the output losses are negligible, the impedance of the crystal tank is Y 2 lf2 Y 2 Z,. = ^ = -- ■! ■ = k^ PI 1—336 (6) Reg Reg Equation ( 1 ) can thus be written 1 _ _ 1 _ _ k Rp k PI Zl 1—336 (7) If k is fixed by output considerations, the percent- age effect of R„ upon the loop reactance becomes a function of R,. alone, being a minimum when R„ is a maximum. If R,, is increased without limit, g,„ approaches k/Zi, as a limit, and the greater the PI, the larger will R,. become. In the case of the Pierce oscillator, it will be recalled that a maximum R,, was obtained by a proper choice of k. This optimum k was the one that provided the maximum excitation voltage. In the Miller circuit, the excitation is the voltage developed across the crystal unit, and thus is limited by the crystal specifications regardless of the value of k. The smaller that k is made, the smaller will be the effective R,„ but, even so, the percentage effect of R„ upon the effective loop reactance will also be smaller. With k fixed by the requirement to eliminate an amplifier stage, the problem of ob- taining a maximum R,, becomes one of keeping the load requirements to a minimum, selecting the vacuum tube, determining the proper operating voltages consistent with the crystal specifications, designing a test model accordingly, and experi- menting for optimum results over the resistance range to be expected in the crystal units. OPERATING CONDITIONS OF MILLER OSCILLATOR PROVIDING MAXIMUM R, FOR GIVEN g„. 1-337. If the bias of a tube is supplied by age, the excitation voltage is small by comparison, so that the operating point of the tube can be theoretically estimated by consulting the R,, and g,„ curves plotted against grid voltage. The operating bias for a given plate voltage w ould approximately be that giving values of R,, and g,„ that obey equation 1 — 336 (7). Unfortunately, there are no curves available that indicate the effective R,. and g,„ for large excitation voltages where the tube is cut off a large fraction of each cycle. Nor has a theoretical basis been established for estimating the probable rates of change in R,, and g,„ as the excitation is increased under various circuit conditions. If time permits, experiments designed to furnish such data may gain for the engineer a valuable insight into the characteristics of his design models. Most probably the “dynamic” curves of R,, and g,„ will correspond closely to the static curves. Yet the possibility exists that significant differences in the rates of change in the tube parameters may be discovered under certain operating conditions. In equation 1 — 336 (7) it can be seen that for any large value of R,„ g„, very nearly equals k/Z,.. For example, if R,, = 0.5 megohm, the difference be- tween g„, and k/Z,, is only 2 micromhos. An R,, of 1 megohm corresponds to practically the same value of g„„ the difference being only on the order of 1 micromho. Thus, when R,, is large, g,„ can be con- sidered more or less a circuit constant. During the time that the tube is cut off, R,, is infinite and g,„ WADC TR 56-156 192 is zero. From the point of view of a large effective Rp it is desirable that the cutoff angle be a maxi- mum. The larger the g„ of the tube above cutoff, the greater can be the cutoff angle. A sharp-cutoff tube would be preferred for this purpose. It is also desirable to have R,, as high as possible above cut- off. For this purpose, a high-mu tube is to be pre- ferred. A theoretical estimate of the optimum relation between the values of Rp and gm, for a tube of the same class-A mu, that provides a maxi- mum over-all effective R„ cannot be attempted here. However, it would seem that the emphasis should be placed upon the larger gm/Rp ratio. The effective Rp of a pentode can always be increased artificially by inserting a high resistance in the plate circuit in series with the oscillating tank, as illustrated in figure 1-142. In testing a given tube for those bias and excitation conditions which pro- vide a maximum Rp, it may be preferable to control the bias independently of the oscillations, or by using an adjustable, r-f-bypassed cathode resistor. A crystal unit should be employed having param- eters known to remain constant over the experi- mental drive-level range. After the tube has warmed up, if the cathode bias is used, the cathode resistance can be decreased until oscillations begin. The cathode resistance can then be increased until the frequency is a maximum. The maximum fre- quency would be an indication of an equilibrium point of maximum Rp. In order for oscillations to be maintained in the event that all the bias is de- veloped across the cathode resistance and the ex- citation is insufficient to drive the grid positive, a Section i Crystal Oscillators small percentage decrease in the excitation ampli- tude must cause at least an equal percentage de- crease in the average plate current, and hence in the bias. Such operation will require that the tube be cut off for a large fraction of each cycle. An adjustable cathode resistance cannot be considered a particularly practical design feature, but it may prove advantageous in an experimental circuit for finding the operating conditions that provide a maximum Rp for a given g„,. FREQUENCY-STABILITY EQUATIONS FOR MILLER CIRCUIT 1-338. Regardless of whether the circuit condi- tions are such that the effective load capacitance of the crystal unit is assumed to be given by equa- tion 1 — 332 (1), by equation 1 — 332 (6), or by equation 1 — 332 (8), the fractional change in fre- quency for a small change in any one of the equiva- lent component capacitances is given by the gen- eral equation df 1 dC, f Fx, ■ C, 1—338 (1) where dC, is equal to dC^, dC,,,,, dC,, dCxp, or dCxp', and represents an incremental change in any of the component capacitances, and Fx-,. is the frequency- stability coefficient of the crystal unit, equal to 2 Ct’/CCx. (See equation 1 — 243(1).) If a tuning capacitor, C,, is connected across Lp in the plate circuit, and if equations 1 — 332 (1) and (2) are assumed approximately correct, it can be shown ■Figvre 1-142. The large resistante, R,/, connected in plate circuit effectively increases R,, of the tube. This method can be used to improve the frequency stability of a Miller oscillator employing a screen-grid tube and an externally connected feed-back capacitance, C,,/ WADC TR 56-156 193 Sactlon I Cryital Oscillators that for variations in C„, df Cpj 2 n w Cp Rp^ dCp “T “ ■ Fxe Xp (1 + RpVXp^) ■ Cp 1—338 (2) where <o = angular frequency. If the circuit is operating at maximum activity, in which case Zi, = Xp = Rp, equation (2) becomes df _ _ Cpg n Rp uCp dCp f C. ■ 2Fx. ■ Cp 1—338 (3) Where equations 1 — 332 (6) and (7) can be as- sumed to be approximately correct, it can be shown that a fractional change in the plate reactance, Xp, causes a fractional frequency deviation of df _ (Cx Cpg) dXp ^ ggg f Cp, Fxe Xp ' If no tuning capacitor is provided to shunt the plate coil Lp, equation (4) can be expressed in terms of a fractional change in Lp, thus: ^ ^ _ (Cx - Cp,) dLp 1^338 (5) f Cp, Fxe L., If desired, equations indicating the frequency sta- bility when other parameters are varied can be derived by following a procedure similar to that employed in the analysis of the Pierce circuit. MILLER CIRCUIT AS A SMALL POWER OSCILLATOR 1-339. The ratio of the output power to the input power is given by the equation P„/P, = R,p/Ro where k = Ep/E,, and R,,, and R„ are the resist- ances represented in figure 1-141. In practice, ra- tios of R„. to R„ can be obtained on the order of 4 for crystal units of maximum effective resistance. If k = 5, this would mean a power ratio of 100. A 10-mw crystal unit could thus be used to develop a 1-watt output. Much higher power outputs, of course, can be obtained with crystal units of small values of R,, or of higher power ratings. It cannot be recomended that a crystal be driven beyond its rated power level, but if an exception should ever arise, the Miller circuit will require the least over- drive. If a larger drive level is necessary than can be obtained with Military Standard crystal units, the cognizant military agency should first be con- sulted. It may be that one or more of the crystal manufacturers has available a nonstandard crystal unit with crystal dimensions and mounting suffi- cient to withstand the required drive — perhaps by operating with an overtone mode — without the risk of significant parameter variations. As a final resort, it will be found that most of the Military Standard crystal units can withstand, without shattering, drive levels from 10 to more than 20 times the rated drive. If need be, power outputs greater than 35 watts can be obtained with the Miller circuit, using a beam power tube or a power pentode. It is much easier to obtain a large output from a high-mu than from a low-mu tube for the same crystal drive. Also, it is easier to obtain a large output from, say, a 50-watt tube operated at low efficiency, than from a smaller tube operated at high efficiency. An r-f choke must be used in the grid circuit if large output is to be developed. Furthermore, fixed bias that is sufficient to pre- vent the grid from drawing current must be used, so as to reduce the grid losses to a minimum. The voltage gain of the oscillator, k = Ep/E„ should be as high as possible. With proper design, except that the crystal unit is operating at tolerances greater than those specified for low drive levels, the Miller circuit can be made to drive a pow'er am- plifier of 300 watts or more. Some crystal units can withstand as much as 120 ma r-f current and still be within the safe-operating range as far as shat- tering is concerned. A pressure-mounted unit is generally to be preferred at high drive levels, be- cause of the added protection it offers, and because its greater thermal conductivity permits the gen- erated heat to escape more rapidly. For maximum output, the oscillator must operate into an im- pedance matching the R„ of the tube. If a Miller oscillator is to drive a power amplifier, great care must be taken in neutralizing the feedback from the amplifier, or the crystal may easily be over- driven to the point of shattering — that is, unless the power amplifier is to serve as a multiplier stage, or if a screen-grid tube is used as the amp- lifier tube. The plate supply voltage for the oscil- lator can be as high as 350 to 650 volts, and that of the power amplifier, 1500 to 2000 volts. If a low- power (7.5 watts, approximately) oscillator tube is used, a fixed bias of 40 to 60 volts will be required for high efficiency. A 3.5 to 4.5-ampere current in the plate L,.C„ tank can be obtained under these conditions. A fixed bias is usually not necessary when a 50-watt tube operated at low efficiency is used. With the same plate voltage as for the low- power tube, a plate tank current of 4.5 to 7.5 am- WAOC TR 56-156 194 peres can be had. When a fixed bias is used, some arrangement must be provided to cut it in after oscillations build up and the negative peaks of Ep must be sufficient for plate limiting to occur at the positive peaks of Ep. When used as a power oscil- lator, the Miller circuit is often required to operate as a variable-tuned circuit, with a coil or tank- circuit in place of the crystal, the circuit thereby being converted into a tuned-plate-tuned-grid or a Hartley type oscillator. Because of this, the vari- ous tuning adjustments and meters that are needed in the variable circuit are also available in the crystal circuit. In this event, the rated load capacitance of the crystal unit will usually exist more in theory than in application. Since the crys- tal is intended to be operated at high drive, the risk is greatly increased that a chance adjustment may overload the crystal to the shattering point. This risk can be minimized by the use of an r-f milliammeter in series with the crystal, with the danger zone well marked. Besides excessive plate voltage and stray feedback due to poor shielding or neutralization, a poorly bypassed screen-grid circuit can lead to an overloaded crystal, as also can an excessive control-grid bias. Now, no attempt should be made to design a circuit in which a crys- tal unit is to be operated above its rated power level unless weight, space, or expense requirements demand the elimination of every possible amplifier stage; unless greater frequency stability is re- quired than can be obtained with a conventional inductor-capacitor network ; or unless a long oper- ating lifetime is not a primary consideration. Even so, if a Military Standard unit is operated beyond specifications, it should be well understood that it is no longer effectively a standard type, and no guarantee exists concerning the replacement of one crystal unit by another. Section I Crystal Oscillators TYPICAL CHARACTERISTICS OF MILLER OSCILLATOR 1-340. Figure 1-143 shows an experimental Miller circuit, the performance characteristics of which were investigated by Messrs. E. A. Roberts, Paul Goldsmith, E. K. Novak, and J. Kurinsky of the Armour Research Foundation at the Illinois In- stitute of Technology. The crystal units used are of the type CR-18/U. The crystal PI indicated for each of the characteristic curves (figures 1-144 to 1-148) of the oscillator in figure 1-143 is the value observed when the crystal was operating into its rated load capacitance. The PI at the rated load capacitance indicates the relative activity of the crystal unit, but is not intended to imply that the same PI is in effect for all variations of the load capacitance. The curves in figure 1-144 indicate (excitation voltage)* and the output voltage as the plate tuning capacitance is varied. An increase in the plate capacitance means an increase in the ef- fective value of L„, so that the frequency decreases. Thus, as Cp increases (Ca in figure 1-144) , the re- actance, X„ of the crystal unit decreases. Oscilla- tions cease whenever the Q, of the grid circuit becomes too small for the proper phase rotation to take place, or the ratio of Xp/X, becomes too high for the feed-back voltage to be of sufficient amplitude, or the plate tank approaches the paral- lel-resonant state, so that E„ can no longer assume its proper phase, which requires the plate arm to be an inductive reactance smaller in magnitude than the capacitive-feedback reactance. The per- centage points in figure 1-144 refer to percentages of the maximum output voltage that was obtained through variations of the plate tank capacitance alone. The six curves shown represent values for figure 1-143. experimeHtal Miller oeeHIpter whoee charatferletk curvet are plotted In figuret 1-144 ta 1-148 WADC TR 56-156 195 Section I Crystal Oscillators an Eb of 200 volts with a full load of 5000 ohms, and for an Eb of 100 volts with full load and with no load. It is interesting to note that in each of the three pairs of curves the maximum grid volt- age occurs at a smaller load capacitance than that at which the output voltage is a maximum. For each curve where the plate tuning capacitance is the same, it can be approximately assumed that the load capacitance is the same. Also, between the values of C,. = 75 and C„ = 90 iint, it can be as- sumed that the percentage change in Ci is small. Since the PI of the crystal unit is the same where the load capacitance is the same, the excitation- voltage-squared curves indicate the relative crys- tal drive for the different Eb and load conditions. Also note that the two pairs of curves represent- ing full-load conditions coincide fairly closely at their points of equal percentages. This is impor- tant in interpreting the curves in ftgure 1-145, each of which represents 50 per cent output, and hence approximately the same load capacitance and frequency. Exceptions are the lOK curves in figure 1-145, as can be checked by figure 1-146. The performances curves in figure 1-145 are the Miller equivalents of the Pierce curves in figure 1-130. It can be seen that the Miller output is much figure 1-144. (A) Square of exeitation vohaga and (B) rms valum of output voltage vanvs plate tuning capac- itance of experimental Miller oscillator, frequency = 7 me; gridleak resistance = I megohm; and PI of CR-IB/U crystal unit (with lead capacitaoce of 32 = 49 kllehms. f.L = full load conditions (S,000 ohms across /date tank) and N.L. = no load conditions. Percentage points refer to percentages of maximum output voltage obtainable under given lead and d-c plate veftago conditions WADC TR 56-156 196 Saction I Crystal Oscillatera WADC TR 56-156 198 more sensitive to changes in the grid resistance. This is to be expected, since the grid-to-cathode r-f impedance and excitation voltage is much greater in the Miller circuit. A crystal r-f voltage of 2 volts represents an excitation of 2 volts in the Miller circuit, but usually of only 1 volt or less in the Pierce circuit. If the curves in figures 1-130 and 1-145 were plotted against excitation voltage in- stead of crystal driving power they would be much more similar in appearance. 1-341. The frequency curves in figure 1-146 are the Miller equivalents of the Pierce curves in figure 1-129. Note that as R, is decreased the frequency falls, whereas in the Pierce circuit the frequency increases. This is one reason why the Miller circuit becomes so much more frequency sensitive to changes in R, when R, is small. As the Q, of the grid circuit is decrees^ because of a decrease in Section I Crystal Osdllotert R„ the effective Zl across the tube must appear more inductive in order for E„ to shift in the correct direction to compensate for the decreased phase rotation between grid and cathode. For this to occur, the net capacitive reactance of the feed- back arm must increase, which can only come about if the inductive reactance of the crystal unit decreases. Hence, the frequency falls, and in so doing, the of the crystal and grid circuit be- comes smaller still, so that an additional drop in the frequency is necessary to compensate for the decrease in the crysta^Q.. In the meantime, the bias decreases and the 'grid goes positive a larger fraction of the time. This tends to decrease Rp, which contributes even more to the drop in fre- quency. With all these effects adding in the same direction, the large frequency sensitivity of the Miller with changes in the grid resistance is ex- plate tunino capacitance - p. lit ffgvf 1-147. (AJ Square of oxcitation voUogo and (B) rmt valuo of output voltago of oxporlmontal Millor oxclllator vonut plat* tuning capacitance for variou* L/C ratio* of plat* tank. Same eryttal unit as wot utod for curves in figure 1-144 WADC TR 56-156 199 S«ction I Crystal Oscillators sxioA SMU'sgvxnoA xndino WADC TR 56-156 200 Figure 1-748, Output and eryttal drive of experimental Miller and Pierte oseillatora as d-c plate voltage Is varied far trystal units of various frequencies and Pi's. The values of PI assume a rated load capacitance of 32 /i/J. Both oscillators were tuned to provide SO percent (max) E„ and to operate Into a 5000-ohm plate lead plained. The sensitivity is a maximum when is a maximum, for then the frequency is a minimum and the crystal is operating nearest its series>reso- nant state. This fact makes an exception to the rule that the larger the effective C„ the greater the stability. 1-342. Of special interest in the curves of figure 1-146 is the fact that those representing the 50- per-cent-maximum-output adjustment show an in- crease in frequency with an Increase in plate volt- age, whereas the curves representing a maximum output voltage show a decrease in frequency when the plate voltage is increased, even though the same plate voltages are applied in each case. Now, in the Miller oscillator, the frequency increases and decreases in the same direction with Rp. The plate characteristics of the 6C4 tube, the tube be- ing used when the curves in figure 1-146 were plotted, indicate a decrease in R,> as the plate volt- age increases. Thus, we should expect the change in frequency of the (max) E„ curves to be due to the change in Rp caused by the change in plate voltage. On the other hand, the oppositely directed change in frequency of the lower-percentage-Ep curves must be due to an oppositely directed change in Rp brought about by a change in the bias. A re-examination of the crystal voltage curves does indeed show at the SO-per-cent-Ep ad- justment that the grid excitation, and hence the bias, is near the maximum. In figure 1-145, it can be seen that for large values of grid resistance the changes in output voltage due to changes in the plate voltage cause a maximum variation in the crystal drive. The evidence is quite strong that there is an operating region between the oppositely changing frequency curves where the changes in Rp due to changes in En and E,. will annul each other. From an inspection of the crystal voltage curves in figure 1-144 we would guess that such operating points will lie on both sides of the maxi- mum-E„ region. Such a state of operation would be an example of “class-D” operation described in paragraph 1-298. 1-343. The curves shown in figure 1-147 indicate the effect of variations in the L/C ratio of the plate tank circuit obtained by increasing the value of Lj in figure 1-143. Increasing L, increases the im- pedance into which the tube operates, and thus increases the r-f plate voltage. This also has the effect of decreasing the frequency and the react- ance of the crystal unit. For this reason, the crystal voltage does not increase in the same pro- portion as the plate voltage. Much greater stabil- ity is obtained with low L/C ratios, but much greater values of Ep/E„ and hence of power gain. Section I Crystal Oscillalert are to be obtained with large L/C ratios. Figure 1-148 compares output-vs-drive curves for several different frequencies and values of PI when the same crystal units are used in both Miller and Pierce oscillators. Note that in the Pierce circuit the slopes of the curves consistently increase with an increase in PI. In the Miller circuit the tendency is for the slopes to increase with decreasing fre- quency primarily, and secondarily with the PI. This may be due to the fact that the gridleak re- sistance used in the Miller circuit was smaller than that in the Pierce circuit. In any event, the average bias will tend to be less at the lower frequencies, since the grid charge has more time during a cycle to leak off. MODIFICATIONS IN DESIGN OF MILLER OSCILLATOR 1-344. A number of Miller oscillators currently being used in military equipment are illustrated in figures 1-149, 1-150, 1-151, 1-152, and 1-153. The values of the circuit parameters, where available, are given in the accompanying circuit-data charts. None of the crystal units employed in these cir- cuits is now recommended for equipments of new design. Nevertheless, all the circuits shown can be modified in one way'or another and used with cur- rently recommended crystal units which have been tested for parallel resonance. The necessary modi- fications would be those that would ensure a cor- rect load capacitance and would not permit a crys- tal to be overdriven within the expected range of effective resistance. The circuits illustrated sug- gest the wide adaptability of the Miller oscillator for different output requirements and uses. It is not possible to single out a particular circuit and declare this design to be preferred. The engineer will need to design and test his own circuit for the particular requirements of the equipment in which his oscillator is to be used. Quite often the type of vacuum tube or other circuit components most readily available influence the design. Unlike the Pierce, the Miller circuit must include a means of adjusting the plate impedance to ensure the cor- rect load capacitance for the crystal if the oscilla- tor is to operate to more than one frequency. In the circuits of figures 1-149 to 1-153, the switching arrangements of those circuits designed to operate over a wide frequency range are for the most part omitted. Most often, a separate plate coil is pro- vided for each crystal position. Because of space limitations in the circuit-data charts, occasionally two different components in a circuit having the same value or being of the same type are assigned the same symbol number. WADC TR 56-156 201 Saclien I Crystal Otcillalers C3 ,, Figure 1-149. Convantlonal Miller otcillator* using triodes Fig. Equipment Purpose F, Fs CR Ri Rj (A) Radio Modulator and Transmitter BC-925 M.O. 556- 740 H Bliley AR-3 100 2 (B) mm II DC-ll-( ), DC-16. DC- 26, or CR- 1 ( )/AR ISO ■ Target Control Transmitting Equipment RC«56>A TOT 6585^ 6167 mm 100 100 TMt Set TS-67/ARN-6 “ 6.9-mc and 20.7- mc signals for testing reocivers in ILS m !■ 50 5 Radio Set AN /ARC-1 A inmii FT 1.6 mniiii Oscillator for hetero- dyne freq meter and crystal calibrator 1000 nFi Dallon's Laborato- ries D-IOOO ■1 15 1 Calibration for vfo of signal generator 5o5o nFi Holder FT-243 82 a I i M.O. 51 > Circuit Data for Figure 1-149. F in lie. R in kilohnu. C in ii/ii. L in fib. WADC TR 56-156 202 WADC TR 56-156 203 S«ctlon I Ciytlal Ofcillotorf FIgun 1-150. ModUtd Mttimr oscMIoten uting triodm$ Fig. Equipment Purpose F, Fs F, Fs CR R. EB E9 (A) Radio Receiver BC-738-DM Ixicel oscillator 6688- 6746 F, 3F| 9Fi Holder FT-243 ■ 180 8.2 220 (B) Radio Receivers R-57/ARN-5 and R-89 ( ) /ARN-5A 6498- 6548 H 1.5 ■ H (C) iiiiiim li B |mM 66 12 H H (D) Nautical- mile rao|(e- synehroniz- ing oscillator 80.86 H II H E^^Qrai^l ■ 8.2 i Circuit Data for Figure 1-150. F in kc. R in kilohma. C in aai. L in ah. WADC TR 56-156 204 WADC TR 56-156 205 Section I Crystal Oscillators (SI figure I -15 1. Miller estlllator* using screen-grM tubes. (C0ectlve suppressor In beam-power tubes Is tonoeeted to eathodo inside tube, although external eoanettloa may be Indicated In diagram.) Fig. Equipment Purpose F, Ft CR Ri Rs Rs R4 R, R« m Monitor ID-18/ CPN-2 Mile range synchro- nizing (Mcillator 98.109 F, 76*C oven 270 47 1 560 0.15 ■ (B) Radio Trans- mitters BC-d3»- 2000- 4000 ■ Holder FT-164 ■ ■ ■ ■ ■ i H Range Marker Generator TD-42/ FPS-8 Nauti- cal-mile range calibra- tor osc m ■ ^^1 i H (D) Radio Trans- mitters BC-640- A,-B,-D M.O. 6566.5- 8666.6 100 0.06 50 85 ■ 1 Radio Trans- mitter T-171B/FR 125- 625 ■ Holder PT-249 5000 1 l66 100 B Circuit Data (or Figure 1-151. F in fcc. R in kilohms. C in lutt. L in mI>. WADC TR 56-156 206 WAOC TR 56-156 Mction I Crystal Oscillators TO AMPLIFIER C«6) ONLY] C(F) ONLY] AUDIO I (NOT USED > WITH crystal) SRI ' ANTENNA hr) only] 500 KC C(F) ONLY] TO ANTENNA CIRCUIT INCANOESCENT lamp, 6'Sv o.is amp C (F) ONLY 3 OF USE IN 500‘KC CIRCUIT harmonic FILTER FI (5<S.t MC) T ( )/AR ^ 12 >10 ~75 flpctro l-ISf. Courtmwd Fig. 1 Equipment Purpose (F) * Radio Set AN /CRT-3 ‘*G)bson Girl” power 06 C Circuit Data for Figure 1-151. F in kc. R in kilohms. C in uttf. L in ph. WAOC TR 56-156 208 WADC TR S6-15« Section I Crystal Oscillators V2 (S) figure 1-752. floctron-eoup/od Milltr oscillaton. All circuits except fXi, (B), and (C) arm Irl-tmt madlUeatlonB Fig. Equipment Purpose F, Fr CR Ri Ra Rs Re Rs (A) Radio Re- ceiver and Se- lector BC- 617-AZ Hetero- dyne oscil- lator F, 4F, CR-1( )/AR 56 2.2 2.2 ■ (B) Radio Transmitters T-3/CRN-2 and T-3A/ CRN-2 HI 6203.7- 6159.26 1 ■ H CR-IA/AR or DC-17.B n^mi ■ H ■ Radio Transmitter BC-329-J M.O. m H 15 il ■■ m (D) Radio Re- ceiver R-146A/ ARW-35 1st het- erodyne oscillator H 6Fi + 5000 100 4f66 (E) Radio Sets AN/TRC-1, -lA, -IB, -1C Hetero- dyne osc in Radio Receiver R-19( )/ TRC-1 7300- 8750 Wi NA CIU/U ^ 100 “155 ■■^55 6 (F) ■1, Hetero- dyne oec in Radio Receiver R-19( )/ TRC-1 7566- 876« 5Fi and 6Fi NA. CR-6/W 100 100 ' 556 M ' (G) Radio Receiver R-19H/ TRC-1 Hetero- dyne oscil- lator 7800- 8750 n ^1^1 100 H H Circuit Oats lor Figure 1-152. F in kc. R in kilohms. C in mJ- L in ^h. NA; Not Applicable. WADC TR 56-156 210 WADC TR 56-156 211 Circuit Data for Figure 1-152. P in kc. R in kilohms. C in /tiJ. L in /ih. NA: Not Applicable. WAOC TR 56-156 212 Stdion I Ciytlal Ofcillcrtert figur* 1-153 Mlse«lfan«ew( Millmr^idllator modHItatlont Purpose ID Fs Fs CR Ri Rs Rs Rs Rs Rs Rr Rs (A) Galvin Radio Trans- mitter PA-8218 P/O AN/ CRC-3 M.O. for phase modula- tor cir- cuit 937.5 1250 F, Fi (mod.) Motorola FMT 1 470 1 10 47 47 0.1 1 (B) Radio Trans- mitter T-264/ FRC-6A M.O. for phase modulator circuit 937 5 1250 F, F, (mod.) 1 470 1 10 47 47 0.1 1 Galvin Radio Trans- mitter PA-8244 P/0 AN/ CRC-3 M.O. for phase modula- tor cir- cuit 3750- 5000 F, F, (mod.) Holder FT-243 1 470 1 47 47 1 0.1 1 (D) Galvin Radio Trans- mitter PA-8026 P/0 AN/ VRC-2 M.O. lor phase modula- tor cir- cuit 3750- 5000 F, F, (mod.) Holder FT-248 1 470 1 47 47 1 0.1 1 Circuit Data for Figure 1-163, F in kc. R in kilohms. C in itid. L in m 1>> unless otherwise noted. 214 WADC TR 56-156 R» Cl Cj C, Cl C. i C, C7 Cs C« Li u , 1 L, Vi V, V, 5000 25 50 j 5-44 5000 10 100 100 2000 1 1 1 7C7 7A8 5600 24 1 6-45 5600 24 100 100 2200 1 1 7C7 7A8 5000 5-44 6000 100 100 2000 5 2000 5 1 R-P choke 1 7C7 7A8 5000 5-44 5000 100 100 2000 5 2000 5 1 R-F choke 1 7C7 1 WADC TR 56-156 215 Circuit Data for Figure 1-153. F in kc. R in kilohma. C in jiid. L in fik, unless otherwise noted. WADC TR 56-156 216 WADC TR 56-156 217 Section I Crystal Oscillators Two-Tube Parallel-Resenonf Crystal Oscillators I-S-IS. Most of the two-tube oscillator circuits are designed so that the crystal is operated at or near its series-resonance frequency. An exception is the multivibrator type of circuit in which the crystal unit is connected between the grid and cathode of one of the tubes. The basic circuit is shown in fig- ure 1-154 (A). If there were no capacitive effects to consider, would be 180 degrees out of phase with Ek, and the 180-degree feed-back inversion would be accomplished entirely by Vj. In this case, the input impedance of V, would have to be purely resistive — that is, the crystal unit would operate at parallel resonance with the input capacitance. The proper phase of Eg, could also be obtained with the crystal unit operating near series reso- nance, and unless the V, input impedance were so reduced that oscillations could not be sustained under such condition there would always be the risk that the oscillator would jump from one equi- librium state to the other. In an actual circuit, the circuit capacitances will prevent V, from operat- ing into a purely resistive load, so that E,„ slightly lags the equivalent generator voltage — j*Eg,. The larger the values of R„ R,,„ Rgg, C„ and C,,, the nearer will the lag in E,„ approach the 90-degree limit. If it is assumed that Rp, is very large com- pared with the reactance of the Vg input capaci- tance, Cgz, the phase of Egj is approximately the same as the phase of E,„. Also, if the ratio, C,/Cgj, is very large, the magnitudes of E*, and Ep, are very nearly equal. As can be seen in figure 1-154 (B), the lag in E,„ causes the equivalent generator voltage of Vg, equal to — /*Eg 2 , to lag Eg,. The cir- cuit will oscillate at that frequency at which the crystal impedance creates the necessary phase dif- ference between Eg, and — E*,. Note, that except for the possible coupling between the output cir- cuits of V, and V- because of the grid-to-plate ca- pacitance of Vg (which can be made negligible by the use of screen-grid tubes), the phase of E,„ and hence of — /lEg^, is entirely independent of im- pedance changes in the V- plate circuit. Thus, to predetermine the angle 9 in figure 1-154 (B), it is only necessary to consider the V, stage as a con- ventional vacuum-tube amplifier circuit. In turn, the V, stage can be treated separately as an equiv- alent circuit driven by a generator of voltage — )nEg 2 . The design must be such that when the crystal unit is operating at its rated X,,, the voltage across the crystal unit differs from the equivalent generator voltage, — /lEg.,, by the desired angle 9 . If Eg, leads E,.,, as indicated in figure 1-154 (B), the feed-back current through C, will be very nearly in phase with E,,,,. The input impedance of V, will appear somewhat inductive, and close to series resonance with C,. The smaller that Cj is made, the higher will be the frequency. If the plate circuit of V, is made inductive, E,,, (not by chang- ing 9 ) can be shifted to be more nearly in phase with Eg, : however, such operation would tend to become unduly critical. For example, assume that E „2 were rotated to where it was in phase with Eg,. This would mean that the phase of the feed-back current, Ig„ with respect to E ,,2 v.as equal to its phase with respect to E,.,, and this in turn would require that the over-all Q of Zg, and Cs in series be the same as the Q of Zg, alone — ^an impossibility unless C, is infinite. But C, cannot be made large without the risk that the circuit will operate as an RC controlled multivibrator. Thus, in the cir- cuit of figure 1-154, E,., must lag Eg, if oscillations are to be maintained. The smaller the phase dif- ference between Ep, and Eg,, the less will be the leading component of Ig, with respect to Ep,, and B+ B + WADC TR 56-156 21S the less will be the leadinsr (inductive) component of E», with respect to I*,. Hence, the smaller the angle e, the more nearly must the crystal approach parallel resonance with the input capacitance of V,. Any change in the V, plate circuit that tends to decrease 9 therefore tends to raise the fre- quency. C*, can be increased by the insertion of a fixed capacitance to make the total approach the rated load capacitance of the crystal unit. 1-346. Since the crystal unit is effectively a ca- pacitance at all frequencies except those near its points of mechanical resonance, some precaution must be taken in the circuit design to ensure that the crystal maintains control over the frequency and that no danger exists that the two-stage cir- cuit can perform as a free-running multivibrator with the frequency controlled by the RC constants. When Ca in figure 1-154 is small and is large by comparison, and when the plate impedances R, and Rb are small by comparison with the R„ of the tube, the feed-back voltage at the low frequen- cies corresponding to the RC time constants can quite easily be kept below the requirements for sustained oscillations. 1-347. No data is available concerning the relative frequency stability of the parallel-resonant multi- vibrator type oscillator, but, from qualitative con- siderations only, it would seem that a performance equal to, and very possibly superior to, that of the average Pierce circuit could be expected, although the operation of the circuit would certainly be much more critical. The V, amplifier stage can be designed to operate into a practically purely re- sistive load, so that fluctuations in the V, plate resistance will have little or no effect upon the frequency. On the other hand, under these condi- tions 9 will be slightly negative and — will lead E,, unless a large reactance, or, preferably, a resistance, is connected in series with C,. If 9 is not critically small, the external circuit of V, can also be designed to appear as a pure resistance, and any variations in the plate resistance of the tube will have a negligible effect on frequency. The annulling of the stray-capacitance effects in the output circuits of the two tubes will require the use of coils, which may not be desirable if a wide frequency range is intended. The phase-shifting Qf of the feed-back circuit is computed in the same manner as in the Pierce and Miller cir- cuits except that the required phase shift is much smaller. 1-348. In the design of such a circuit, the value of 9 can be arbitrarily predetermined. Assume that when R, of the crystal unit is a maximum, the feed-back current, I,,, is to be in phase with — /lE*,. Section I Crystal Oscillators With R», assumed to be large compared with the average PI of the crystal unit, estimate the capaci- tance C», required across the crystal unit for the Q of Z„, to equal the tangent of 9 when the X, of the crystal unit corresponds to the reactance of the rated load capacitance. Make Cj such that its re- actance at the operating frequency is equal to the reactive component of Z^, computed above. Under these conditions Ig, will lag — for all values of R.. less than maximum, since the Q of Zg, would be greater than tan 9 if the frequency did not in- crease to bring Zg, nearer its antiresonant value. But the increase in the frequency as a result of a decrease in R,, is less than that which would occur if the circuit were designed so that Ig, woulc' lead — ^lEg- for most values of R,.. Also, if the feed-back circuit is designed for series resonance when R, is a maximum, the values of C-, should prove more practical, the operation will be less critical, and there is the assurance that all values of R,. will permit oscillation. The plate-circuit impedances of the two tubes are next determined so that the feed-back voltage is sufficient to maintain oscilla- tions when R, of the crystal unit is a maximum. Generally, neither tube should operate into a load exceeding 5000 ohms. The plate voltages are chosen so that the crystal unit cannot be driven beyond the rated drive level. With the phase char- acteristics of the circuit determined, more or less by design, to ensure a proper load capacitance for the crystal, it may be that the optimum operating voltage will be more readily determined through experiment. At equilibrium, the total voltage gain of the loop, from Eg, to E,„ to Eg^ to E,,. and back to Eg,, must be equal to unity. Thus, G, G2 Ga G, E,i 2 Eg 2 Eg. ■ Ep, Ep2 Eg2 Eg. _ j Ep2 1—348 (1) where, referring to the circuit in figure 1—154, G. = Ep,/E„=^ =^g„.Rg Kpi -T ^pi 1—348 (2) Gj = Egj/Ep, = 1 ' 1-348 (3) Gs = Ep2/Eg2 = ^ >-1 = g:n2 Rb Rp2 "T ^p2 1—348 (4) Gg = Eg,/Ep2 = 1 1-348 (5) So G. G 2 Gs G 4 = gn,l gm 2 R» Rb = 1 1—348 (6) WADC TR 56-156 219 Section I Crystal Oscillators Equation (6) is only a first-order approximation in which it is assumed the plate resistances of the tubes are very large compared with the external plate impedances, Z„, and which, in turn, are approximately equal to R, and Rb, respectively. Also, it is assumed that X,., is small compared with Zfi, and that either X,., is small compared with Zj, or the tendency towards a series-resonant rise in voltage across Zg, is sufficient to make equation (5) approximately correct. 1-349. The output is most often taken from across a 500- to 1000-ohm resistance between the cathode of Vi and ground. In this event, R.^ connects di- rectly to the cathode of Vj — not to ground. The cathode output is quite useful for matching to low- impedance inputs, such as would occur, for ex- ample, when feeding a coaxial line. Regardless of where the output is obtained, it can be seen that its amplitude cannot be expected to greatly exceed that of the Pierce circuit. Since two amplifier stages are required and no additional gain is pro- duced, there can be little advantage in using the multivibrator circuit unless thermostatic control of the temperature is employed and the design is such as to ensure greater frequency stability than can be achieved in the Pierce circuit. The use of a single dual-type tube offers greatest economy. The principal advantages of the circuit are its relative independence of fluctuations in the tube voltage, and its adaptability for impedance-match- ing to low-impedance output circuits, 1-350. Figure 1-155 illustrates three multivibra- tor-type crystal oscillators that were designed for use in Diversity Receiving Equipment AN/ FRR-3 ( ). Circuit (A) is a later-model replace- ment of circuit (B). Very possibly the preference for (A) is at least partly due to a desire to elimi- nate the variable effects of the inductors in (B) with changes in frequency. From the data avail- able it cannot be said that the crystal in (B) is not actually operating at or very near its series- resonance frequency. The state of operation of the R4 figure 1-155. 7wo-$tag» paralM-n$onant o*elMaton of muhivibrator typ*. Fig. Equipment Purpose Pi CR Ri Rt R> R4 Rs R« Rt (A) Diversity Receiving Equipment AN/FRR-3A Local oscillator 1400- 3800 Holder FT-249 250 100 5 ■ 0.6 0.16 ■ (B) Diversity Receiving Equipment AN/FRR-3 Local 08C. 1400- 3800 Holder FT-249 (Entire circuit in 65* C oven) 250 i ■ ■ 1 ■ ■ ■ 1 Diversity Receiving Equipment AN/FRR-3A 6f5 with AFC react- ance tube, V. 462.45 50 mii^ 10 mill ■ I 6 1 Circuit Data for Figure 1-155. F in kc. R in kilohma. C in ii/it. L in ^h. WADC TR 56-156 220 SccKen I Cryitol OtciilcrtArt R5 flgur* I- 155. Continued R« C, C: Cj C, Cs C. Ct Cs L, V 1 -V 2 Vj 80 10 100 2000 6SN7GT 6AC7 10 200 75 50 miiiiii 100 50,000 100 ■ m ■ 0.6 100,000 ■ 250 50 10 100 2000 2200 6SN7GT 6V6GT WADC TR 56-156 221 SacHon I Cryttql OKiilaton crystal will vary quite widely with changes in the tuning of the LjCa tank. Circuit (C) is a beat- frequency oscillator which is crystal-stabilized when switch S, is in the crystal position, as shown. The triode-connected beam power tube serves as a reactance tube, which effectively shunts the crystal with a capacitance that varies with the bias supplied to the control grid of the tube. The bias, in turn, is controlled by the a-f-c discrimi- nator circuit in a teletype terminal. The purpose of the circuit is to ensure that the beat frequency remains constant even though the frequency of the incoming signal should vary slightly. If the beat frequency tends to drift, the sign and magnitude of the discriminator output causes the bias of the reactance tube to effectively change the load ca- pacitance of the crystal unit in such a direction that the frequency of the oscillator rises or falls by approximately the same number of cycles per second as does the incoming signal. Oscillators with Crystals Having Two Sots of Eloctredos 1-351. The original crystal oscillator devised by Dr. Nicolson, as well as a number of the earlier crystal oscillators tested by Dr. Cady, employed crystals with, effectively, two pairs of electrodes. The basic circuit is shown in figure 1-156. The re- quired phase inversion of the amplifier output voltage is provided by the crystal unit operating at a mode for which the polarities of the plate and grid terminals with respect to ground are 180 degrees out of phase. The circuit shown operates the crystal unit very near its series-resonance fre- quency. In practice, a capacitor is normally con- nected between crystal and ground, so that the circuit is more commonly employed for parallel- mode tested crystal units. Still, it is not without some license that we classify this type of oscillator as a parallel-mode type. The crystals most appli- cable for this class of circuit are the very-low- frequency elements of the X group, which vibrate in lengthwise extensional or flexural modes. The electrode connections that permit the desired phase inversion depend upon the particular crystal element. Assume that electrodes numbers 1 and 3 are on one side of the crystal, and that 2 and 4 are on the opposite side, as indicated in figure 1-156 (A). For a flexure element, such as element N, where electrodes 1 and 3 parallel each other down the length of the crystal, as shown in figure 1-156 (B), the flexure mode is excited when the potential across 1 and 2 is oppositely polarized to that across 3 and 4. If the same electrode arrange- ment is to be used to excite an extentional mode (or the flexural mode of the duplex element J) the polarities of the two sets of electrodes must be in phase. In this case, the connections of one set of electrodes should be reversed in the circuit shown in figure 1-156 (A). For example, plates 2 and 3 should be connected to ground and plate 4 should be connected to the grid, if the proper phase inversion is to be obtained. A crystal having the two sets of electrodes at opposite ends of the crys- tal, as shown in figure 1-156 (C), would be driven at the second harmonic of the length extensional mode (or of the flexural mode of a duplex crystal) , if connected as shown in figure 1-156 (A). Greater stability and a smaller crystal arc possible for a given frequency by operating at the fundamental mode. To permit this, if the crystal unit is plated as shown in figure 1-156 (C), the connections of one pair of electrodes should be the reverse of those shown in figrure 1-156 (A). If it can be assumed that the current in the grid circuit is negligible compared with the crystal current between ter- minals 1 and 2, and if the stray capacitance be- tween the two sets of electrodes is ignored, the equivalent circuit between terminals 1 and 2 will appear approximately as shown in figure 1-156 (D). L, C, and Co represent the parameters of a fully plated crystal. A more exact analysis of this type of crystal unit can be found in the book “Electromechanical Transducers and Wave Fil- ters” by W. P. Mason, D. Van Nostrand Co. 1-352. Figure 1-157 shows a practical oscillator design employing crystal units having two sets of electrodes. Although the electrode connections shown for CR would indicate that the plate and 2 <81 (C) (0) figure 1-156. Basic circuit ot oseillator using crystal with two pairs of elaetrodos WADC TR 56-156 222 SacHon i Cryctal Oicillatort flgun t-IS7. Practical crystal-oscillator dmsign omploying vory-low-frsquoitcy crystal unit -with two sols of oloctrodos grid terminals always connect to the same side of the crystal unit, the actual connections will depend upon the particular element used. The variable capacitor Ci permits a frequency adjustment of approximately 60 parts per million. To ensure that the crystal unit is operating into its rated load capacitance, the exact frequency of a test crystal should be known when it is at series resonance with its rated capacitance. To a first approxima- tion, the terminal that is to be connected to the grid can be assumed to be open-circuited, so that the resonance to be tested is that between the rated Cl and one half of the crystal. With the test crystal connected in the oscillator circuit, C, can be ad- justed to provide an output at the previously measured “rated” frequency of the test crystal. This adjustment therefore will provide the rated load capacitance for all crystal units of the same type. The varistor is inserted to protect the crystal unit from overdrive, and to ensure a stable output voltage. As recommended by Bell Telephone Lab- oratory engineers, the nominal values of R„ C,, C„ and C .1 that provide satisfactory operation in the 1 . 2 - to 10 -kc frequency range are given in the following table. The values shown will provide a direct current in M of approximately 12 micro- amperes. Frequency Range (kc) R. (kil- ohms) c. (w^f) c, (mt) c, (M/if) 1.2 — 1.5 100 1.5 — 2.0 100 Hia 2.0 — 2.5 100 160 500 2.6 — 3.2 100 160 600 3.2 -- 4.5 100 120 500 Frequency Range (kc) R. (kil- ohms) c, (mm^) c, (mm^) C 3 (/*Mf) 4.5 — 6.7 100 120 700 250 6.7 — 8.0 100 90 500 250 8.0 — 10.0 51 90 1000 0 1-353. Figure 1-158 shows the crystal oscillator in Test Set TS-251/UP, which employs a duplex crystal element. The crystal circuit is used to syn- chronize the blocking oscillator at a frequency of 1818.18 cps. The output of the blocking oscillator is for counting down to 303.03 pulses per second, which, in turn, are used in checking Loran pulse- repetition rates. A CR-ll/U crystal unit is used which has a resonant frequency of 1817.44 ±0.3 cps at 75° Fahrenheit. The rated maximum effec- tive resistance of the crystal is 30,000 ohms, and its rated maximum permissible current is 0.03 milliampere. The fixed capacitance paralleling the variable capacitance is used only if necessary. The varistor is rated at 1 ma/14V for temperatures between 75 and 86 degrees Fahrenheit. Crystal and Ma9ic-Ey« Rasenaace Indicator 1-354. An interesting application of a parallel- resonant crystal circuit is the tuning indicator shown in figure 1-159. When the tuned frequency, F„ of a variable oscillator is equal to the antireso- nant frequency of the crystal unit in parallel with the input capacitance of V,, a magic-eye tube, the excitation of V, is a maximum, as is the current through R,, and, hence, also the shadow angle of the indicator. The circuit thus provides a con- stant visual crystal check on the tuned oscillator frequency. Different crystal units can be switched in for different channels. WADC TR 56-156 223 Sactlen I Crystal Oscillators Figure 1-158. Oup/ex-ofoctrodo irystal eireult in Tost Sot T5-251/UP for synchronizing blocking oKlIlator at I SI 8.18 cps. Resistanco is given in kllohms, capacitance in micromicrofaradt Figure 1-159. Crystal and magic-eye resonance indicator Fig. Equipment F, CR . R, Ra (A) Radio Transmitter BC-696-A 3000- DC-8-C.-, 5 1 0 39 4000 D,-K (B) Radio Transmitter BC-457-A 4000- DC-8-C,- 10 0.39 5300 D,-K (C) Radio Transmitter BC-458-A 5300- DC-8-C. 15 0,39 7000 D,-K (D) Radio Transmitter BC-45d-A 7000- DC-8-C. 5.1 0.39 9100 D,-K Circuit Data for Figure 1-159. F in kc. R in kilohms. C in mmI- 224 WADC TR 56-156 S*cH»n I Crystal Oscillators SHIES-RBSONANT CRYSTAL OSCILLATORS 1-355. For maximum frequency stability it is gen- erally preferable to operate a crystal unit at its series-resonance frequency, but series-mode cir- cuits are most widely used for overtone operation. At series resonance the crystal element appears as a resistance, so that in the normal circuit it can be short-circuited or replaced by a comparable resist- ance without stopping oscillations. Series-resonant oscillators generally have smaller outputs than do oscillators of the parallel-resonant type. Also, series-resonant oscillators usually require more circuit components, and hence are not often used except in the very-high-frequency range. In gen- eral, the operation of the series-mode circuits is less complicated than that of the parallel-mode oscillators. Nevertheless, the circuit design be- comes incrjeasingly critical at the higher frequen- cies and higher overtones. The stray capacitances must be kept tc a minimum, and all leads must be as short as possible. It may be necessary to nullify the crystal shunt capacitance, Co, by connecting across the crystal unit an inductor that is anti- resonant with C„ at the operating frequency. It may also be desirable to connect a capacitor in series with the crystal unit, to tune out the stray inductance of the crystal leads. Tuned circuits must be provided if a crystal unit is to be driven at a particular overtone mode. Quite often, satis- factory operation is obtained simply by designing a conventional variable-tuned oscillator to operate at the desired frequency, and then inserting the crystal unit in a plate tank or feed-back circuit. There will be a range of tuning adjustments in which the crystal can assume control and hold the frequency very nearly constant. As the tuning ad- justments are varied beyond this range, the con- trol becomes quite unstable or ceases altogether. Usually, the region of stable control becomes smaller as the overtone order is increased. If broad-band operation is desired with no tuning ad- justment other than the selector switch for chang- ing the crystal, additional precautions must be taken to ensure that oscillations cannot be main- tained except when the crystal impedance is small — that is, the crystal unit is operating near series resonance. For maximum frequency stability, the effective resistance of the circuit facing the crys- tal unit should be as small as possible. At the higher frequencies, the stray capacitances limit the impedances obtainable from the tuned circuits, thereby making them more selective and hence more effective in influencing the frequency and in increasing the instability. 1-356. The series-mode oscillators most widely recommended are listed in the following table, and rated according to their relative design and per- formance characteristics. A rating of 1 represents the top relative superiority in the corresponding characteristic. It should be understood that the ratings are based upon average qualitative results which might well be contradicted by the data of individual investigators. Any one of the series- mode circuits expertly designed could surpass the performance of a poorly designed circuit rated higher in a particular characteristic. The fre- quency-stability rating assumes average oven and voltage regulation. Rs R4 Rs Rs C, V, 1.5 1000 51 0.02 50,000 1629 1.0 1000 51 0.02 50,000 1629 1.0 1000 51 0.02 50,000 1629 1.5 1000 51 0.02 50,000 1629 WADC TR 56-156 225 Section I Crystal Oscillators Symbol of Oscillator A B C D E F G H I B K Bi Frequency Stability (%) 0.0001 p b 0 to 0.0005 0.0004 0.0004 0.0002 0.0015 0.002 0.002 0.0015 Power Output 6 5 3 1 3 5 4 4 4 2 1 3 Versatility 4 4 2 1 2 3 2 3 8 3 2 3 Upper Frequency Level 5 1 3 2 2 2 2 3 3 3 4 3 High-Resistance Crystals 2 6 3 4 4 4 2 4 4 4 1 3 Ease of Adjustment 6 7 1 3 2 2 2 5 5 5 4 5 Untuned Bandwidth 5 5 2 2 1 2 1 6 6 6 4 6 Frequency Multiplication 5 5 2 4 4 1 5 5 3 3 6 6 Low Harmonic Output 1 2 2 5 6 5 5 4 4 4 3 2 Circuit Simplicity 4 5 3 2 a 2 2 3 3 3 4 4 Isolation from Load 1 2 3 4 B 2 4 4 4 4 4 4 Low-Frequency Operation 1 6 3 6 5 5 2 4 4 4 1 2 The oscillator symbols in the foregoing table cor- respond to the respective index letters of the oscillators listed below. Names of Series-Mode Oscillators A. Meacham Bridge B. Capacitance Bridge C. Butler, or Cathode-Coupled D. Grounded-Cathode, Transformer-Coupled Type E. Grounded-Grid, Transformer-Coupled Type F. Grounded-Plate, Transformer-Coupled Type G. Transitron H. Impedance-Inverting Transitron I. Impedance-Inverting Pierce J. Impedance-Inverting Miller K. Grounded-Cathode Two-Stage Feedback L. Modified Colpitts, C.I. Meter Type Meacham Bridge Oscillator 1-357. The Meacham bridge oscillator, illustrated in figure 1-160, provides the greatest frequency stability of any vacuum-tube oscillator yet de- vised, but the region of maximum frequency sta- bility is limited to the lower frequencies because of the increased effect of the stray circuit capaci- tances when the frequency becomes greater than a few hundred kilocycles per second. The oscillator is of the crystal-stabilized type employing tuned circuits. At frequencies above 1000 kc the effect of the stray capacitance is sufficient to reduce the stability to a point where little is to be gained by the use of the Meacham circuit. The oscillator is principally employed with GT-cut crystals in fre- quency standards, to generate frequencies of 100 kc. In figure 1-160, it can be seen that if the bridge were perfectly balanced there would be no excita- tion voltage. At the start of oscillations, the ratio of Ri (practically equal to the series-arm R of the crystal unit) to R, is smaller than the ratio of the Rj to Ri. But R4 is a thermistor — it is the resistance of a tungsten lamp which sharply increases in value as the temperature rises. (A semiconductor such as carbon, silicon, or germa- nium can be used, in which case the resistance will decrease with temperature. The negative tempera- ture coefficient of the semiconductor is generally larger than the positive coefficient of tungsten, but the semiconductor thermistor is more expensive and is much more difficult to duplicate because of its great sensitivity to impurities.) As oscillations build up, the current through Rj increases to a point where the heat generated from the power losses raises the the thermistor temperature, and hence the resistance, to a point where the bridge is almost balanced. Equilibrium is reached when the imbalance of the bridge is just sufficient to sup- ply heat to the thermistor at the same rate at which it escapes. For maximum amplitude stabil- ity, the ambient temperature of Ri should not be permitted to vary over a wide range. Normally, the tungsten lamp will heat to a dull red of approxi- mately 600 degrees centigrade. A variation of over 100 degrees in the ambient temperature could have a significant effect on tlie equilibrium power losses in the thermistor, if extreme precision were de- sired. The operating temperature of the lamp is very low compared with the rated temperature, and consequently the lamp can be expected to last indefinitely. The oscillator should be designed and adjusted so that the phase shift occurs entirely in the bridge. That is, the tube should operate into WADC TR 56-156 226 SecHon I Crystal Oscillators Hgun 1-160. Ba*k cirtuit of Moaeham bridge-stabilized oscillator a pure resistance, so that the instant the plate current is maximum the peak transformer volt- ages should occur with polarities as indicated in figure 1-160. Transformers having powdered-iron, toroidal cores can provide a coefficient of coupling very close to unity in the low-frequency range. The following analysis of the frequency stability and the activity stability of the Meacham oscilla- tor, except for minor deviations and extensions, has been guided by the postulates and basic con- siderations as presented by W. A. Edson.* FREQUENCY STABILITY OF MEACHAM BRIDGE OSCILLATOR 1-358. First, we shall assume that the vacuum tube in figure 1-160 operates into a purely resistive load, and that the entire phase reversal takes place in the bridge transformer. The phasor diagram in figure 1-161 (A) shows the relation of the volt- age EIo to the other voltages of the bridge network. (Refer to figure 1-160 for voltage symbols.) Next, assume that some change in the capacitance of the circuit requires that Eo be shifted in phase by a very small angle equal to but that the change is so small that the magnitude of all the bridge voltages can be assumed to remain constant. In order to produce the phase shift it can be seen that E„ and hence the current through R„ must be rotated by an angle e. In a triangle with angles A and B opposite to sides a and b, respectively, * Vacuum Tu6e Oscillators, John Wiley and Sons, 1953. WADC TR 56-156 o ][^ — : — T- = ^ (Law of Sines). Likewise, in the sin A sin B triangle EoEjE,, Eo/sin d = E 4 /sin <t> or sin 6 = sin 0 1 — 358 (1) E4 Since we are assuming that both d and 6 are very small, equation (1) can be written, approximately, e 1—358 (2) E4 (B) Figure 1-161. Phasor diagrams of bridge voltages in Meacham oscillator. Angles 6 9 (greatly magni- fied) represent small shifts in phase when crystal unit is operating slightly off series resonance 227 Section I Crystal OKillatert The current through the input transformer can be assumed to be negligible compared with the total current through the crystal, so the current through Rj is essentially the same as that through the crystal. Under these conditions, as is shown in paragraph 1-241, 6 ~ tan d Xe Ri + R2 2LAu R -|- R 2 1—358 (3) where L and R are series-arm parameters of the crystal unit. By rearranging equation (3) and di- viding by <u, the fractional change in frequency required to produce an angle B is found to be A« 0 (R -h R 2 ) e R„VC , ,,, IT = 2a,L = -z7L~ where Rc = R -f R, is the total resistance of the crystal side of the bridge, and C is the series-arm capacitance. On substitution of equation (2) in equation (4), it is found that Ao? 0 Eo Rc / C o> 2E4 \ L It now remains to determine the magnitude of for a small change in the capacitance of the cir- cuit. The most likely changes in capacitance take place in the grid circuit, the average aC* usually being on the order of 10 times the average ACp in the plate circuit. Looking away from the grid, it can be seen that when the bridge is very nearly balanced, the grid faces a resistive impedance N/ (Ri -f- R3) (R2 + R4) Ri "f" R2 "i" R3 "I" R4 1—358 (9) The capacitance Cg will have been adjusted to be effectively antiresonant with the leakage induct- ance of the transformer, which inductance can be imagined to be in parallel with R*. Still looking away from the grid, we can imagine a generator connected between grid and cathode. If the capaci- tance should change by a small amount aC», the ratio of the excess reactive component of current to the resistive component becomes R,<oACg. This will equal the tangent of the phase shift, <i>, which is sufficiently small for tan to be assumed to equal ip. Thus, with 4 , = R,<uAC(, on substituting equation (9) for Rg, equation (8) becomes Equation (5) indicates that the more nearly bal- anced the bridge (the smaller the Ep/E* ratio), the greater will be the frequency stability. Now, ^ Letting (R3 -f- R4)/R4 = (Ri + R^l/Ra = m. we see that E0/E4 = • E, Also, since Aw _ / R2 Ra — Ri R4 \ (Ri Ra) (R2 ~1~ R4) <0 \ R2 R4 / (Ri -j- Ra + Ra + R4) Re N,^ ACg 2L 1—358 (10) Now, let us assume that Ri and Rg remain fixed. As Ri is varied, R, must vary in direct proportion to keep the bridge balanced. If R, = kR„ R4 will always approximately equal kR,. Substituting these values of R, and R« in all terms where the error introduced can be considered negligible, equa- tion (10) becomes E = E 51—^ '^•VR8 + R4 R. + R2y ^ Ri* (kRa - R4) (1 4- k) N,* AC, 0) kRa 2L 1-358 (11) where Rp = Ra + R4, then If R4 is expressed as being equal to R, (k — i), equation (11) becomes E0/E4 = m(^-|i) = Ra/R4 - Ri/Ra On substitution in equation (5), we have Aw w / Ra R3 — Ri R4 \ Re ^ I, RaR4 /2VI7C 1—858 (8) ^ ^ (1 -I- k) i Ri'* Ng’* ACg w “ 2kL 1—858 (12) 1-869. Equation 1 — 358 (12) does not quite indi- cate the relations among all the circuit parameters that are effective in providing an optimum fre- WADC TR 56-156 228 quency stability. It is first necessary to determine how i, which is a measure of the imbalance in the circuit, is dependent upon the other parameters. For this purpose, it is necessary to find that value of i which must exist in order for the feed-back voltage to be at equilibrium. It will be assumed that the r-f plate current. Ip, is equal to gmEg. If this assumption is not warranted, g„ in any of the following equations can be replaced by /i/ (Rp -|- Zp) . To a first approximation, Zp = Np* R. Rt/(Rc + Rt) 1—359 (1) and El — Ep/Np — Ip Zp/Np — g„ Eg Np Rp Rx/ (Ro + Rt) 1—369 (2) also, Eg = NgEp, and E„ is given by equation 1 — 358(7). On substitution in equation (2) 1 = go. Np Ng .^Rc Ra - Ri Rt)/(R. + Rt) 1—359 (3) Equation (3) is the equilibrium feed-back equa- tion for the Meacham oscillator. On expressing Ra and Ri as functions of R, and R^, it is found that at equilibrium (1 + k) Rp gn. Np Ng Ri Rj 1—359 (4) Substituting (4) in equation 1 — 358 (12) ^ = (1 + k)^ Ri Rp Ng AC, « 2 k g,p Rj Np L 1—359 (5) Section I Crystal OsciUaters Also, caution must be taken that in improving the stability in one respect, it is not impaired to a greater extent in another. Since the expected ACg is approximately 10 times the expected ACp, the ratio of the right-hand sides of equations (5) and (6) can be equated to 1, with 10 aCp substituted for aCg. On thus dividing (5) by (6) ^ 10 (1 + k)^ Ri Ra k Rp' Np' or .t 2 ,xt 2 10(l-|-k)'R. 10 (1 + k)^ (m - 1) " kmRp "■ km' 1—359 (7) With the oscillator designed according to equation (7) , average capacitance variations in the plate and grid circuits will have approximately equal effects upon the frequency. When the square root of equation (7) is combined with equations (5) and (6), and Rp is expressed as mR,/(m — 1), Aw _ ACg m' Ri j 1 -|- k w 2 L g,„ V 10 k (m — 1) 1—359 (8) and ^ / lO (1 + k) w 2Lg^ \ k (m - 1) 1—359 (9) If the oscillator is to be designed on the basis of equation (7), it can be seen that k should be made as large as is practicable, and m should be such that the factor mVVn^ — 1 is a minimum. It can be shown that this occurs when In a similar manner, in equating to ZpuACp, it can be shown that for small changes in the plate capacitance ^ = R.^ Np ACp o) 2 (1 + k) g,p R/ Ng L 1—359 (6) It can be seen that greater stability is to be had when gn, is a maximum and when the ratio m = R«/Rt is small. For changes in Cg, the optimum value of k is 1 (when (1 -f k)Vk passes through a minimum) . For changes in Cp, it would be desir- able to have k as large as practicable. A further consideration is to so proportion the parameters that the expected variations in C, and Cp will have the maximum opportunity to cancel in their effects. m = 4/3 1—359 (10) Frequency stability, of course, is not the only con- sideration; there are also the vacuum-tube and thermistor characteristics and the power rating of the crystal unit that must be taken into account in deciding upon the optimum parameter rela- tions. Remember, that in equations (4), (5), and (6), gm can be replaced by the more exact term /i/(Rp -f Zp). Certainly, a high-mu tube is to be preferred, and when it is operated class A the sec- ond-harmonic output can be expected to be at least 65 db below the fundamental. The screen voltage should be fairly high, in order to increase gm. Nor- mal operating voltages can be employed, but Eg should not be allowed to drive the grid positive. WADC TR 56-156 229 Saction I Crystal Oscillators ACTIVITY STABILITY OF MEACHAM BRIDGE OSCILLATOR 1-360. Starting with Eo, the input to the grid transformer, we see that at equilibrium the prod- uct of the gains of all the stages, from Eo back to Eo, must be equal to 1. Thus, Gain: Voltage: Gi G2 Ga G t — Eo G, N. X X 1/N, X i ^ i ^ i = E,/Eo^ g Gz = Ep/E,^ j, Ga = E,/Ep^ ^ 1 E. From equation 1 — 358(12) it can be seen that in the interest of frequency stability, i, and hence the imbalance of the bridge should be as small as pos- sible. Fortunately, this condition also agrees with the requirements of high activity stability, for the smaller the difference of the actual thermistor re- sistance from a value equal to kR:, the larger will be the percentage change in that difference for a small change in the thermistor voltage. Equation 1 — 358(7) can be written Eo = G 4 E. = i R. R2 E,/k Ro" 1^60 (1) where i = (kRj — R4)/R2 and G4 is the gain of the stage. In the over-all gain equation, above, Gj and Gj can be considered constant, so that G« primarily has the function of compensating any changes in G2 of the vacuum tube. From equation (1), it can be seen that iE,/Eo can be considered a constant. Or, in the over-all gain equation we see that Gi G2 G3 i R, R2/k Re* = 1 1—360 (2) or that If the effects on E, due to the changes in E. and i exactly cancel so that dEo = 0, then, by equations (3) and (5) — dG2/G2 “ di/i = — dEe/Eg Under these circumstances it can be seen that the percentage change in the activity is exactly equal to the change in the gain of the tube. If the ther- mistor is to be effective in preventing the ampli- tude of the output from changing significantly with changes in G2, clearly an increase in E, must produce a decrease in E„. We can define the activ- ity sensitivity of the bridge to be s E, di/i dE, 1—360 (6) The sensitivity is thus defined as the percentage variation in i per percentage variation in the volt- age across the bridge. The problem now is to con- vert equation (6) into a function (equation 14) of the circuit constants so that s can be predeter- mined by the design engineer. From equations (5) and (6) we find that the percentage change in E„ per percentage change in E, is G2 i = k Rc* Np/Ri R2 Ng = constant On differentiating, i dG2 -h G2 di = 0 di dG 2 i G 2 1—360 (3) Equation (3) shows that for a given percentage change in the gain of the tube, the smaller the value of i, the smaller need be the change in R, to restore equilibrium. Since G2 ^ gmZ,„ we can write di/i = - dg„/g,„ 1-360 (4) On differentiating iE,/Eo = C, where C is a con- stant, we find that i dE, -f E. di - C dEo = 0 1—360 (5) E, dEo E„ dE. s — 1 1—360 (7) In practice, E„/E, (^ G,) can be on the order of 0.003 or smaller; so, if the change in Eo is com- parable to that of E, in magnitude, excellent am- plitude stability will be achieved. The stability depends first upon the magnitude of i, and sec- ondly, upon the sensitivity of the thermistor. The latter is defined as E 4 dR^ _ d(log R 4 ) R4 dE4 d(iog E4) 1—360 ( 8 ) Figure 1-162 shows the resistance-voltage char- acteristics of a number of tungsten lamps for ambient temperatures at room values. For lower ambient temperatures, the curves would be shifted to the right somewhat, and for higher tempera- tures, to the left. Since the curves are plotted on log paper, according to equation (8) it can be seen WADC TR 56-156 230 0.03 0.10 0.3 1.0 3 10 30 APPLiEO POTENTIAL -VOLTS FIgun Ui62. katiatancm of typical fungatan lamps vnut appllad veltaga and pawar dtsalpatlon whan tha amblant tamparatvn la 300° Kalvin acala (approxtmatmly 27° C) that the thennistor sensitivity S at a griven value of E 4 is the actual slope of a curve at that point. It is important, of course, to operate the thermis- tor at a voltage where the slope approaches a maxi- mum. It is convenient to express the bridge s as a function of the thermistor S. O' ■ ^ 4L Since 1 = 5 , then Its di = or dR 4 _ k dR 4 R 2 R 4 1---360 (9) E.di ^ k E. dR 4 ^ kS / E.dE 4 \ i dE, i R 4 dE, i \ E 4 dE, / 1-360 (10) Now, E4 = E2 — Eo = Ea, but dE4 ( = dE* — dEo) is not approximately equal to dEa. So dE4/E4 == (dEa — dEo)/Ea = (dE, — mdEo)/E„ where m = R E ^ On substitution for dE 4 /E 4 in equation Xv2 ^2 ( 10 ), we find rearranging, we have Section I Crystal Oscillators kSE, - mkSEo iE, — mkSE„ 1—360 (12) The term mkSE„ in the numerator can be consid- ered negligible, and dropped. After expressing E, in terms of equation ( 1 ) and rearranging, it is found that s = kSRc/i(Rc - SRj) 1—360 (13) Finally, on substituting for i its value given by equation 1 — 359(4), we are able to express the ac- tivity sensitivity entirely in terms of the known circuit parameters. Thus, ^ kSg„.NpN,RiRa ® (1 -h k) (R. - SR.) or 1—360 (14) s = kSg^NpN^R./d + k)[m- S(m - 1)] The reciprocal of s can be considered the percent- age gain in the output voltage (or in Ep or E.) for a unit percentage change in the gain of the tube, since dGa/G, is equal to — di/i. In the equation for s, note that if the thermistor sensitivity were equal to R,./Ri, the stability mathematically would be infinite. Since Rc/R. is greater than 1 , a single tungsten lamp could not provide the thermistor sensitivity for the above condition to hold unless special measures were available to reduce the heat leakage from the filament. The effective sensitivity could be increased if R, were replaced by another tungsten lamp, and the crystal unit were inserted in the place of R,. Theoretically, the sensitivity can be made much larger than unity simply by vary- ing the ambient temperature together with the op- erating temperature of the filament ; for instance, by constructing a thermistor with the filament mounted inside a heater sleeve and controlling the heater current by feedback from a later amplifier stage. If equation (14) is taken apart, it will be found that the denominator term, (Rc — SR.), originates from that component of dE, that is equal to — dE„. When there is an increase in E„ the voltage E, changes in two ways: one is due to the change in the current through R,, and the other is due to the increase in the resistance, itself. It is the latter component that is approximately equal to — dE,. Mathematically, the change in E, is expressed by the differential equation By equation (7) we see that dE,/dE. = E„(l — s)/E.. On substitution in (11) and after 1 350 (15) WADC TR 56-156 231 Section I Crystal Oscillators Since dR4/R4 is equal to SdE4/E4, on substitution in equation (15) it can be shown that dR4 = S 1-360 (16) If S is greater than 1, an increase in voltage across R4 must result in a decrease in current. (Inciden- tally, since the change in R4 is actually due to a change in the temperature brought about by an increase in power, a value of S greater than unity implies that the percentage increase in resistance is at least equal to twice the percentage decrease in current.) Now, assuming that the cujrrent through the input transformer is negligible, E, = I4RT, and dE, = l4dR4 -f RTdl4, where dRi = dRi. If E, is to remain constant, that is, if dE, is to equal zero for a small change in the gain of the dR tube, dh/U must equal — If the latter value Xv^ is substituted in equation (16), it will be found that for conditions of s = 00 : S = R*!* Rt — R4 Rt _ Rc R3 Ri 1—360 (17) This is the explanation of the term (Rc — SRi) in the denominator of equation (14). Other than the assumption that the changes in the current I» through the grid transformer can be considered negligible in their effect upon dE4, the term is enticely a function of the Rj and R4 arms of the bridge, and is not related to the gain characteris- tics of the rest of the circuit. No experimental data is available concerning the operation of the Meacham bridge oscillator with values of S greater than unity, when R, behaves as a negative resist- ance (an increase in Ej is accompanied by a de- crease in I4). Theoretically, if S were greater than Rc/Ri, an increase in the gn, of the tube would ultimately result in a decrease in the output- volt- age and in the voltage applied across the bridge. In an actual circuit, whether stable values of R4 would be maintained under such conditions is open to question. Perhaps the thermal lag of the fila- ment and the extreme sensitivity of E* would so influence the operation that R, would periodically overshoot its mark and prevent an unmodulated equilibrium from being reached. In practice, the values of S will be on the order 0.5, so such con- siderations do not arise. For s to be as large as possible, referring to equation (14), it can be seen that [k/ (1 -f- k) ] should be as large as practicable. This agrees with the equations for frequency sta- bility if the circuit is to be designed according to equation 1 — 359(7). The term [k/(l -j- k)] has no maximum, but approaches unity as k increases indefinitely. Assume that s is equal to 50. This means that a change in the gain of the tube of 1 per cent will cause a change of only one-fiftieth of 1 per cent in the output voltage. Or in terms of db, since s = E. dE. di i A (log i) A(log E.) A(log Gt) Adb in tube gain A(log Ep) Adb in output an increase of 0.5 db in the gain of the tube will cause only a 0.01-db increase in the output. CRYSTAL DRIVE LEVEL CONSIDERATIONS IN MEACHAM BRIDGE OSCILLATOR 1-361. A starting consideration in the design of a Meacham bridge oscillator is that the crystal unit is not to be overdriven. If Pj is the crystal power, Ic = VPi/Ri is the crystal current, and mRi / Pi m-y/ Pi R, E, I- Rn ^ UL - -i\ Ri m 1—361 (1) With Ri determined by . the crystal unit, it is de- sirable, from the point of view of frequency sta- bility, for m, and hence Rz to have small optimum values. If S approaches unity, the small Rj will also be an important consideration in activity sta- bility, but for normal values of S the activity stability is improved slightly if Rj is large. The (ro - SR,) ' in equation 1 — 360 (14) has no term . ,Ro - SR,. maximum, but approaches unity as Rj is increased indefinitely and R, and S are held constant. Usu- ally, the requirements of frequency stability are the more important, and Rz should be kept as small as practical thermistor resistances and values of k permit. At low frequencies, values of R, may be in the neighborhood of 1000 ohms or more. The voltage across the thermistor will be T?, E, _ le Rj _ E] _ ■%/ Pi Ri n;4 — ; ^ ; — m m — 1 m — 1 m — 1 1—361 (2) where Ei is the voltage across the crystal unit. For convenience, we repeat equation 1 — 360 (1), but ex- pressed as a function of m and k: WADC TR 56-T56 233 Ep = (m — 1) i E,/km“ 1—361 (3) section I Crystql Oscillators The power dissipation in R 4 is P« = £414 = E, I./(m - 1) k = P,/(m - 1) k (max) E 4 = ~\/ Pcm Pm (m — l)\/ir 1—362 (1) 1—361 (4) The impedance of the bridge in terms of R, is Z. = R. Rt/(Rc + Rt) = kmRi/(l + k) (m - 1 ) 1—361 (5) The plate impedance of the tube is Zp = Np» Z. = The plate voltage is Np* kmR, (H-k)(m-l) 1—361 ( 6 ) E. = Ip Zp = 8 m Np^kmRi ” " (1 + k) (m - 1 ) (7) Also, Ep = Np E. = mNpVPT^ /(m - 1) 1—361 ( 8 ) and E. = N, Eo 1—361 (9) Finally, we repeat equation 1 — 359 (3), the over-all equation for feed-back equilibrium, but expressed in tmns of Ri, m, and k: G, G2 G, G4 = 8m Np i Ri ^ m (1 •+■ k) Rj can be adjusted to provide the same value of m for each different crystal unit. Under these cir- cumstances, E, and Ep will be the same in each oscillator, even though Ri varies. Two fundamental problems are that the design must ensure that the crystal current does not overdrive the crystal unit when R, is small, and that the thermistor current is sufficient for S to be a maximum. DESIGN PROCEDURE FOR MEACHAM BRIDGE OSCILLATOR 1-S62. The fixed point of reference for estimating the current and voltage at any point in the Meaeham circuit is the thermistor voltage E^. This is the voltage that is required to make R* = R,/(m — 1). If R, and m are held constant, Et as well as E, (= mEt) and E, ( = NpE,) will also be constant. If Pnn is the rated crystsd power, Rm is the maximum series resistance of the crystal I> unit, and is the minimum expected resistance of the crys^ unit, then, by equation 1 — 361 (2), E 4 must not be greater than the value Since the Meaeham oscillator is most applicable for use in the low-frequency range where crystal units having resistances in the neighborhood one or more thousand ohms are not uncommon, the risk is greatly increased that an exceptionally well-mounted crystal will have a resistance of as little, as, perhaps, Rm/25. Also, since the Meaeham circuit is primarily useful as a precision oscilla- tor, an additional safety factor should be allowed to prevent the crystal unit from being driven be- yond its test specifications. For these reasons, it is suggested that in the absence of prior experi- ence or manufacturer’s recommendations for a given type of crystal unit, the Meaeham design for frequencies below 200 kc assume a minimum R of Rm/25, rather than Rm/9 as was assumed in the case of the parallel-resonant oscillator design. However, it can still be assumed that the most probable crystal unit will have an R = Rm/8. If the crystal unit to be used is a precision GT cut, a safety factor as large as N = 25 neW not be made. In any event, crystal units having resist- ances less than Rm/9 can be expected to be ex- tremely rare, and if N = 9 is considered a suffi- cient safety factor, an output voltage two-thirds greater can be realized than if N is assumed to be 25. A crystal unit having a resistance less than Rin/9 would be driven beyond its test level, but far below a level that could damage the crystal. Since the resistance is already low, an increase in resistance with overdrive would do more good than harm. The only concern is that the frequency of a borderline crystal may deviate beyond the tolerance limits. Such a risk could be checked dur- ing a production test, but would subtract from the reliability of crystal replacements in the field. In equation 1 — 361 (10), it can be seen that when k is a minimum (when R, = R„), the imbalance, as measured by i, is a minimum. When k is large, the percentage changes in (k 4 - 1 ) and in Ri are very nearly equal, so that the imbalance tends to vary as the square of R,. k should be chosen for maximum frequency stability under variations of C„ assuming that the crystal unit resistance is its most probable value (approximately Rm/3). Ac- cording to equation 1 — 359 (5), with all else fixed, the percentage change in frequency is a minimum when k = 1. The most probable optimum value of k, therefore, fixes R, as equal to Rm/3. Thus, for any random value of R„ k = Rm/3R, 1—862 (2) WADC TR M-156 233 Section I Cryttal Oscillators Next, a value of m equal to 4/3 (see equation 1 — 359 (10) ) should be chosen, to provide maxi- mum frequency stability on the assumption that equations 1 — 359 (8) and (9) are to apply when R, is its most probable value. After this is done, a safety factor of N should be selected, and the maximum value of E4 should be determined by equation (1) , such that it will not require a bridge voltage sufficient to overdrive the crystal unit when R, is equal to R.„/N. Next, a thermistor is R chosen that will provide a value of R, = ^ m — 1 when Ej is equal to, or less than, the maximum , value determined above. Next, the ratio Np/N, can be determined, using equation 1—359 (7) with the assumption that k = 1 and m = 4/3. This gives N a ratio ^ = \/T5 «= 4, which value thus provides the greatest probability that random changes in Cp and C,, can cancel when R, is its most probable value. The next step is to select a tube with hig^i class-A gm and R„. A 6AC7 would be very satisfac- tory. Using equation 1 — 361 (6), determine Np on R R the assumption that Z„ = when R, = Now, Np can be made equal to N|,/4. R;, of course, must be variable over a percentage range comparable to that to be expected from the crystal unit. Normal tube voltages are used. The other circuit compo- nents can be determined according to the tube specifications for class-A operation and the special output requirements of the oscillator. Ep, E„ E„, Ip, I,., etc can be determined from the equations in paragraph 1-361, the frequency stability from equations 1 — 359 (5) and (6), and the activity stability from equation 1 — 360 (14) for maxi- mum, most probable, and minimum values of R,. MODIFICATIONS OF MEACHAM BRIDGE DESIGN 1-363. Two designs of the Meacham bridge sta- bilized oscillator are shown in figure 1-163. In each Fig. Equipment Purpoae F, CR (A) Control-Monitor IP-68/CPN-2A Timing osc. Con- trola indicator sweep freq and prr of shoran station 186.22 Oven controlled (B) Radio Set AN/FRC-10 Carrier osc and phase-shift cir- cuit 100 WECo D-163897 D-169649 Ri Rs Rs Rt 0.1 Thermi- stor 250 0.12 , Circuit Data for Figure 1-163. F in kc. R in kilohms. C in iiiti. h in jih. 234 WADC TR 56-1S6 ! ■< * Section I Oyticil Oscilloton of these circuits inductor-capacitor combination has been connected in series with the crystal unit. Obviously the combination is intended to be reso- nant at the crystal frequency. The variable ar- rangement shown in figure 1-163 (A) permits the frequency to be pulled to a more exact value if desired, the crystal unit (if necessary) operating with a reactive component in its impedance. Or, in case the tube operates into a partly reactive load, the tuning elements in the bridge could permit the crystal, itself, to operate at exactly series reso- nance. The series inductor and capacitor are effec- tive in aiding the initial build up of oscillations and in ensuring that the crystal assumes control at the frequency of the desired mode. It can also be presumed that the LC combination in the bridge improves the waveform somewhat and reduces the small distortion introduced by the tungsten lamp. This distortion is due to the fact that the filament cools at least to some extent during the time that the current alternates from its effective value in one direction to its effective value in the opposite direction. At frequencies above 100 cycles per sec- ond this distortion in the waveform is not serious. At radio frequencies it is normally small compared with the distortion introduced by the tube. The resistance R; in figure 1-163 (A) appears to be inserted in order to maintain a constant tube load by minimizing the variations in the bridge im- pedance due to adjustments and to crystal units having difference resistances. In figure 1-163 (B) note that the crystal unit is grounded. This is the usual arrangement. The parallel primary wind- ings of the grid transformer in the same figure suggest that the arrangement is an adaption of a readily available transformer, very probably of the same construction as the one in the plate cir- cuit. The parallel primary connection is in the di- rection of phase addition. Because the near-unity coupling between the coils effectively doubles the R. Rt c, c . c, c. c» c ^ VI 22 ■ 0.56 1 no 40 300 100,000 10,000 300 10,000 6AU6 2651 0.15 0.072 ■ 140 140 100 100 500,000 500,000 500,000 WADC TR 56-156 235 Section I Crystal Oscillator* inductance of each, the parallel connection pro- vides the same step-up arrangement and primary impedance that would be provided by only one of the coils if used alone, but with a reduction in the leakage inductance. A Meacham bridge-stabilized oscillator can be designed employing two or more tubes. On the average, slightly better frequency stability can be achieved with a two-tube circuit, but only in rare instances are the additional cost, space, and weight requirements worth the small improvement in performance. Perhaps, at fre- quencies in the neighborhood 1000 kc or higher the two-tube arrangement could be more profitable than the one-tube stage. The design of a multi- stage bridge oscillator can be practically the same as that of the one-stage circuit except that the tube gain, Gj, is replaced by G 21 G 2 J . . . Gs„, where Gzk is the voltage gain of a transformation stage between the output and input of the bridge, and where n is the total number of such transforma- tions. By increasing the number of positive-db stages, the bridge i can be made as small as de- sired, and the frequency and activity stability will be increased in proportion to the gain. It is because the possible gain is unlimited for all practical pur- poses that the Meacham oscillator represents the ultimate in precision control of the frequency. In the final analysis the limiting condition is the de- gree to which the crystal parameters, themselves, can be kept constant. Figure 1-164 shows the basic circuit of a two-tube Meacham oscillator that em- ploys no transformers and offers the advantage of only a single tuned stage. Although the design equations are somewhat different from those of the conventional one-tube stage, the same basic approach is to be employed, and the problems to be encountered can be solved similarly to those of the transformer-coupled circuits. Capaeitance-lrid9e Oscillators 1-364. Capacitance-bridge oscillators may po.ssibly prove suitable for use in the v-h-f range. Their advantage lies in the fact that a properly balanced capacitance bridge cannot provide sufficient feed- back of the proper phase to sustain oscillations at any frequency other than the tuned frequency of the circuit, provided a crystal unit is connected in the circuit that has a resonant mode of vibra^ tion at the tuned frequency. A properly balanc^ capacitance-bridge oscillator is thus crystal-con- trolled, rather than crystal-stabilized. On the other • hand, if the bridge is not balanced, the circuit can operate as a free-running oscillator, which may or may not be crystal-stabilized. For the purpose of ensuring operation of crystal units at desig- nated very high harmonic modes, the capacitance bridge, if not the most dependable, is at least as dependable as any other so far tested. The princi- pal disadvantage of this type of circuit is that rather critical tuning adjustments must be made, '^« and one crystal unit cannot replace another unless these adjustments are repeated. Largely on this account the circuit is not to be preferred for fre- quencies below 50 me, and perhaps not below 76/y me. Nevertheless, once the bridge is properly ad-^ justed, the operation with a crystal unit free of spurious modes is dependable under any extremes in temperature that can be reasonably expected. BASIC CIRCUIT OF CAPACITANCE- BRIDGE OSCILLATOR *1-365. Figure 1-165 illustrates the basic circuit * The discussion in paragraph 1-36S is based upon the analysis of the basic circuit app^ring in the report, H.F. Harmonic Crystal Investigation, by S. A. Robinson and F. N. Barry of Philco Corporation, on Army Contract #W3S-038 8C-14172, 1947. WADC TR 56-156 figvn 1-164, Two-slago Moaeftom brUgm-stablllxnd osciMotor 336 B-f Figure 1-165. (A) Basic circuit of capacitance-bridge oscillatar. (Bi Equivalent circuit of capacitance- bridge oscillator of a capacitance-bridge oscillator. The first design consideration in this oscillator, as in all others that are to operate in the v-h-f range, is to keep all the leads as short as possible. Those circuit ar- rangements and circuit components that provide a minimum of stray inductance and capacitance are to be preferred. In the circuit of figure 1-165 (A), Lp and L, are actually a single, tapped induc- tor with the two sides wound on the same form and tightly coupled together. The induced-voltage effect is equivalent to that of a single generator connected across both coils and driving the bridge with an emf (Ep -f E,) Ep(Np -1- 1). Np, the turns ratio of Lp to L„ is usually, and most con- veniently, equal to 1. In case the shunt capacitance of the crystal unit is greater than 10 it would be desirable to make Np slightly greater than 1. An Np greater than 1 but less than 2 can be ex- pected to provide a higher output, but the opera- tion will tend to be more critical and the frequency less stable. Before the circuit is placed in opera- tion, the bridge must be balanced at an off-reso- nance frequency, so that no voltage can appear across the grid circuit. At an off-resonance fre- quency the crystal unit appears as a capacitance Co, so that under the conditions of balance 1-^65 ( 1 ) With Np ^ 1, Cb is adjusted to equal C*. (Cp is here assumed to include Cpp, and Lp to account for Cp. See figure 1-165 (B).) The total capacitance Section I Crystal Oscillators (B) in the circuit is thus. Ct = Cl + 1—365 (2) 2 Since the distributed inductance of the crystal leads, L', tends to increase the effective value of Co, the frequency at which the bridge is balanced should not be greatly different from the intended operating frequency. If L' is unduly large, a series capacitance should be connected in the crystal arm of the bridge sufficient to annul the stray induct- ance in the vicinity of the operating frequency. Once the bridge is balanced, Cp should not be ad- justed again. The initial adjustment is rather critical, requiring an accuracy of a few tenths of a micromicrofarad. Co and Cp in the v-h-f range should be as small as possible. The coaxially- mounted crystals, such as those contained in the HC-IO/U holder, are to be preferred on this ac- count. Values of C„ in the neighborhood of 4 or 5 /ifif are quite feasible. Co can be further reduced by connecting an inductor across the crystal unit to annul part, but not all, of the shunt capacitance ; however, this should be avoided, because the pres- ence of the inductor would narrow the frequency WADC TR 56-156 237 Section I Crystal OKillators range over which the bridge can be considered balanced. Cb must be adjustable over the expected capacitance range of the particular type of crys- tal unit to be used. C, is the tuning capacitance. For crystal-controlled operation, Ci is adjusted so that the total circuit capacitance Ct is approxi- mately resonant with the total inductance at the operating frequency. To balance the circuit, C, is set to a position that tunes the bridge to a fre- quency well off the resonance frequency of the crystal unit. Referring to figure 1-165 (C), it can be seen that E, equals (E, + Ecb)- At the tuned frequency, R can be neglected and the crystal unit considered as a capacitance, C,. Approximately, E, and Ecr are 180 degrees out of phase, and there- fore tend to annul each other. Now, assiune that Cb is made to approach zero. 1 , and Ecr therefore become negligible, and the circuit behaves as if the crystal side of the bridge were open-circuited at Cb. The remaining circuit would be simply a Hartley oscillator with the crystal unit serving as a blocking capacitor between the inductor and the grid. If Cb is gradually increased, Ecr builds up until a point is reached where E, effectively is can- celed and E( is insufficient to sustain oscillations. Cb should then be increased one more increment beyond the oscillation cutoff. At this setting of Cb, the bridge can be considered properly balanced, but a check should first be made that oscillations do not occur at other settings of C, well removed from its value for crystal control. If such oscilla- tions do occur, the adjustment of Cb should be repeated. The free-running oscillations can be dis- tinguished from the crystal-controlled oscillations by the continuous nature of their activity curves as measured by grid current and output meters when Ct is varied above and below a discontinuous region. A discontinuous point indicates an abrupt Ip-r|+l2+l3+l4 figun 1-165. (C> SimpMlod •quivafent circuit change to crystal control, where the frequency be- gins to change at a much slower rate with varia- tions in circuit capacitance. However, once the bridge is balanced, no oscillations occur except near the crystal resonance frequency, in which region the bridge balance is upset. 1-366. Referring again to figure 1-165 (C), with the circuit balanced, suppose that C, is gradually increased from its minimum value. At some point oscillations suddenly start; as Ci is further in- creased, the activity builds up to a maximum and then sharply declines, as is illustrated in figure 1-166. Note also the sharp decrease in frequency when maximum amplitude is approached. Appar- ently, when oscillations first begin, the crystal ap- pears inductive. Ecr therefore has a large compo- nent in phase with E„ and the circuit is essentially a modified Miller oscillator. Also, the ratio of I 4 to I, is a maximum, since Ecr tends to cancel the voltage across Cb. As C, is slowly increased, the frequency and the inductive reactance of the crys- tal drop. This means that the effective Q of the grid circuit also decreases. Although the presence of the capacitance C, modifies the phase relations, the circuit performs fundamentally as a Miller oscillator. L, can be interpreted as something of a booster inductor to increase the effective induct- ance of the crystal unit, and 1 ^ can similarly be viewed as a booster current to boost the voltage across the inductive component in the grid circuit without, at the same time, increasing the voltage across the crystal R*. That the capacitance-bridge circuit actually has the same characteristics as does the Miller circuit is well illustrated by the similarities between the curves of figure 1-166 and the equivalent curves for the Miller oscillator shown in figure 1-144. Note that for both oscilla- tors, the circuit capacitance for maximum excita- tion does not coincide with, but is smaller than, the value for maximum output. One significant difference between the two circuits is the fact that the Miller circuit cannot maintain the proper feed- back phase if the crystal is operated at series resonance, whereas the capacitance-bridge circuit can, because of the presence of L,. Where oscilla- tion cutoff for the Miller circuit is above the series- resonance frequency of the crystal, it is below the series-resonance frequency in the capacitance- bridge circuit. 1-367. If the crystal control is to be fully effective, the series-arm resistance must be small compared with the shunt reactance, Xco- A’thougi. this requirement becomes increa’ingl' diffi lit at the higher harmonics, it te avhievect, uven at frequencies well above lOc m' . Assuming that WADC TR 56-156 238 Section I Crytlal Otcillalon FIgun 1-166. Typical performance curves of capacitance-bridge oscillator, showing effects of change In bridge tuning capacitance on voltage output, activity (d-c grid current), and frequency the series-arm R is not more than one-tenth the magnitude of Xc, then the approximate equation for the effective crystal reactance, X, = X.Xco/(Xco -|- X,), where X, = 4irLAf series-arm reactance, is sufficiently close for an interpretation of the capacitance-bridge performance. Now, os- cillations cannot start unless |Xcb| > X* -t- Xl,. Xcb we shall assume is equal to Xco under the con- ditions of balance. X, is equal to |Xco|, and hence to |Xcb|, when X, = — Xob/2, that is, when the crys- tal unit is halfway between series resonance and antiresonance. Thus, when oscillations start, the crystal frequency must be much nearer to the reso- nant than to the antiresonant state. Also, the plate circuit must appear inductive to the vacuum tube to a degree dependent upon the effective Q of the grid circuit. This means that Ii must be slightly greater than (I* -f- Is + I 4 ). In figure 1-165 (A), it can be seen that the crystal unit operates into a load reactance approximately equal to the paral- lel combination of C, and the inductor (Lp -f L.) in series with Cb. As the reactance of Cj approaches that of the inductor, the reactance of the parallel combination rises very sharply, and a small change in C, can make a large change in the load reactance across the crystal unit. More than any other factor, this is the reason for the sharp dip in the fre- quency curve as C, approaches a maximum. 1-368. It is not possible to tell at which point in the curve the crystal passes through series reso- nance. Since at series resonance the reactance of C, in parallel with L„ and L, is equal, approxi- mately, to — Xcb, the resonance frequency may well be below the knee of the curve for a crystal having a very small Co (conditions for large Xcb and near- parallel resonance of C, with the inductor) and above the knee for crystals of larger At series resonance, if R is small h approximately equals laCb/Ci. Assuming that E.(= I.Xl,) leads Ecb(= RR) by 90 degrees, the effective Q of the grid circuit at series resonance is equal to E,/Eck. When Eo and Ecr are expressed as functions of I 3 , Cb, C„ and Xi,„ it can be shown that (series reso- nance) Qp = ^ = Xi..(C,-^Cb) equation is only a broad approximation in the y-h-f range, since all the distributed parameters have been ignored, particularly the grid capacitance and the resistance of the inductor. However, it does indicate that the larger the ratio of C, to Cb, or, equivalently, Xco/Xci, the smaller the inductive phase shift that will be required in E,„ and the more nearly will the bridge tank approach parallel resonance. If R, or rather the total grid losses, should increase or decrease, the frequency will de- crease or increase, respectively. It seems safe to assume that crystal units having the larger values of RCo products will operate fairly near their WADC TR 56-156 239 Section I Cryctal Otcillolor* series-resonant state. This is due partly to the fact that the smaller the Xoo/R ratio, the smaller the frequency range between resonance and antires- onance. Crystal units having the smaller values of RCo will perform with greater amplitude and frequency stability if operated above series reso- nance. Unfortunately, crystal units in the v-h-f range are tested only for series resonance. The greater likelihood of the occurrence of unwanted modes increases the importance of having the cir- cuit designed so that the operation of the crystal unit lies within its tested specifications. While the capacitance bridge is excellent for preventing all modes of oscillation except the one desired, it is not a true series-mode oscillator, although it is so classified because its v-h-f application requires the use of crystals that are only series-tested. Rather, the oscillator is something of a hybrid between a Miller and a stabilized Hartley circuit. In the interest of frequency and amplitude sta- bility, the circuit should be adjusted to operate above the knee of the frequency curve. A setting of the tuning capacitance corresponding to a grid current of 50 per cent of the maximum possible provides, approximately, the optimum output volt- age and operating state nearest series resonance that are consistent with the operating region of better stability. The peak of the voltage-output curve in figure 1-166 corresponds closely to the adjustment for maximum tank impedance, which certainly occurs below series resonance where the crystal unit appears as a capacitance. The larger the capacitive reactance that the crystal unit can have and still permit oscillations, the more nearly can series-resonance oscillations fall within the higher stability region. For this purpose, the ratio of L„ to Lp and of Ch to C„ should be as large as unity, or greater, when the capacitance-bridge oscillator is to be used with series-tested crystal units. DESIGN MODELS FOR CAPACITANCE- BRIDGE OSCILLATORS 1-369. The circuits shown in figures 1-167 through 1-171 represent five different modifications of the capacitance-bridge oscillator. These circuits were designed and tested by the research team of S. A. Robinson and F. N. Barry of Philco Corporation. No single type of circuit was found to be superior for operation over the entire tested frequency range of 50 to 200 me, but each circuit has advan- tages for certain applications. The inductive arms of the bridge can be a single, self-supporting tapped inductor having an inside diameter of one- quarter inch or greater. Silver-plated AWG No. 16 wire can be used. The tuning and balancing capaci- tances are small air capacitors. The fixed capaci- tances are, for the most part, the button-mica Erie type. Composition resistances are used, having nominal values of ±10 per cent. Successful opera- tion of any of the circuits depends largely upon arranging the circuit components to permit the shortest possible leads, and all components should be of small physical size. Silver-plating of the com- ponents is desirable, and careful shielding and the use of low-loss insulating materials is necessary. Without good shielding and well-insulated capaci- tor shafts, it may be impossible to adjust the bridge properly because of the effects of hand ca- pacitance. Transmit-time effects become quite sig- nificant as the frequency is increased beyond 50 me. The lag in the response of the plate current with rapid changes in the grid voltages is equiva- lent to the circuit behavior that would result if an inductance were connected in series with R,, of the tube. The lower the plate voltage, the larger is the apparent inductance and its accompanying tendency to lower the frequency. Usually, this effect makes it easier to operate the crystal unit at series resonance, but the need for careful B + regulation becomes all the more important. For normal voltages, transit lag is approximately 0.2 degree per megacycle in v-h-f tubes such as the 6AK5. SINGLE-TUBE, 50- TO 90-MC CAPACITANCE-BRIDGE OSCILLATOR 1-370. The circuit shown in figure 1-167 has been operated at frequencies as high as 135 me, but its particular merit lies in its performance at frequen- cies between 50 and 90 me. When operated in the high-stability region, output up to 10 volts can be obtained, although care should be taken that the rated drive level of the crystal is not exceeded. Outputs of 2 milliwatts into an inductively cou- pled 100-ohm resistor can be obtained in the same operating region. COMPACT, MINIATURE, 50- TO 120-MC CAPACITANCE-BRIDGE OSCILLATOR 1-371. The circuit shown in figure 1-168 is par- ticularly suited for construction as a small, pack- aged, plug-in oscillator. If desired, several such oscillators of different frequencies can be designed as interchangeable units of the associated equip- ment. The entire shielded unit need not occupy a space greater than 6 cubic inches. The maximum frequency at which this circuit was found to oscil- late was 156 me, but the activity at that frequency was less than one-tenth that at 50 me. At 120 me the activity is approximately one-fourth of that at WADC TR 56-156 240 Section I Crystal Oscillators Figure 1-167. A eingle-tube capacitance-bridge oscillator which is practical for operation in the 50—90-mc frequency range. Resistors not otherwise specified are Vi w. L-1 and 1-2 are a single center-tapped ceil of suitable inductance Figure 1-168. A plug-in capacitance-bridge oscillater which is practical for 50-150-mc frequency range WADC TR 56-156 241 Section I Crystal Oscillators 50 me, so 120 me appears to be the most praetieal upper frequeney limit. A subminiature tube hav- ing high transeonduetance is used. Greater output is to be aehieved with a triode, but greater fre- quency stability is to be had with a pentode. With a triode, the comparatively large plate-to-grid ca- pacitance which shunts the balancing condenser may make it difficult to achieve a balancing capaci- tance as small as that of the crystal unit. This condition requires that the L-2A section of the bridge inductor be somewhat larger than the L-2B section. The possible output is reduced thereby, but the crystal unit will be operated nearer its series-resonance frequency. The output secondary, L,, can be a single turn coupled to the plate end of L,. CAPACITANCE-BRIDGE OSCILLATOR FOR GREATER POWER OUTPUT IN THE 50- TO 80-MC RANGE 1-372. The circuit shown in figure 1-169 was de- signed for the purpose of achieving a maximum power output without regard to the rated drive level of the crystal unit. Ho'wever, none of the crystals used were fractured during the experi- ments. The higher-power circuit is essentially the same as that of figure 1-167 except that the N,, ratio of the bridge inductor is greater, higher volt- ages are used, and a 6AG7 replaces the 6AG5 tube. Although the 6AG7 has a higher transconductance and power rating than the 6AG5, the interelec- trode capacitances are greater, the internal leads are longer, and the base is constructed of higher- loss material. The circuit operated at frequencies as high as 102 me, but above 80 me the disadvan- tages introduced by the vacuum-tube construction make the circuit impractical. Better performance might be expected with a 6AH6. With the tube operated near its maximum rated dissipation, a one-watt inductively coupled output was obtained at 54 me, and one-third watt at 80 me. These out- puts are representative of the peak obtainable. Much less power is to be had if the oscillator is ad- justed for operation in the higher-stability region. TWO-TUBE, 50- TO 100-MC CAPACITANCE-BRIDGE OSCILLATOR 1-373. The circuit shown in figure 1-170 is similar in operation to the one-tube circuit except that the feedback has an additional antplifier stage to boost the gain. There is a significant difference in that the crystal unit is connected to the plate aide of the bridge. Under this arrangement, the excitation voltage of V, lags the r-f plate voltage of V„ which means that if the plate load is resistive the r-f plate voltage of V, would tend to lag the re- quired excitation voltage of V.,. For oscillations to occur, the plate tuning tank of V, must appear inductive in order to shift the input of V. to the proper phase. After equilibrium is reached, a slight increase in the value of C, causes the plate impedance of V, to become more nearly resistive, and therefore the input of V. becomes more nearly 180 degrees out of phase with the input of V,. This requires that the frequency drop to a point where the voltage across C ^ is more nearly in phase with the r-f plate voltage for V,. For both tubes to operate into resistive loads, the crystal unit must appear as a capacitance. For the crystal unit to operate near series resonance and at the same time maintain the oscillations in the higher-stability region, it would seem that R,, the parasitic damping resistor in the input circuit of V. can be replaced, if necessary, by a resistance comparable in value to the V. input reactance. The effect will be to shift the input phase in a lagging direction, which would require the V, tank to be more detuned, and hence less critically adjusted. This, in turn, will require a comparable shift in the phase of the input to V„ which is to be had by a decrease in frequency, thereby per- mitting the bridge to be less critically tuned in the vicinity of the crystal resonance point. The circuit in figure 1-170 was found quite practical for use as a test oscillator for measuring the relative per- formance characteristics of harmonic-mode crys- tal units. During the temperature runs, even though frost had collected on various components, the operation of the circuit was little affected. For duplicate units of this circuit to provide essentially the same meter readings for tests of the same crystal unit, it is necessary that the vacuum tubes used in the twin circuits show the same plate char- acteristics within ±5 per cent. A breadboard model of the oscillator having different values of tuning inductances was able to operate at 140 me. L„ in figure 1-170, is a 5-turn coil, approximately one-quarter inch in diameter; La and L., are the two halves of a 4-turn, center-tapped coil, approxi- mately one-half inch in diameter. MULTITUBE CAPACITANCE-BRIDGE OSCILLATOR OPERABLE AT FREQUENCIES UP TO AND ABOVE 200 MC 1-374. The circuit shown in figure 1-171 has been used to generate crystal-controlled frequencies as high as 219 me, the seventy-third harmonic of a 3-mc crystal. This frequency approaches the ulti- mate directly obtainable with quartz crystals at the present state of the art. A large part of the WADC TR 56-156 242 Section I Cryetal OKillalon Figure 1-169. A eapaeitanee-bridge oscillator tor higher power output which is practical for operation in the SOSO-mc frequency range. Resistors not otherwise specified are Vi w. All fixed capacitors have mica dieiectrics figure 1-170. A iwo-tube capacitance-bridge oscillator which Is practical for operation in the 50-l00>mc frequency range WADC TR 56-156 243 S«ction I Crytlol Osdilcitor* / Figure 1-171. A multitube capacitance-bridge aeclllator which la practical for operation in the 50-200-mc range. Keaiatora not otherwise specified are rated at 'Aw success of the oscillator in figure 1-171 is due to the balanced electrical and mechanical nature of the push-pull capacitance-bridge circuit. The op- eration is very much the same as that of the cir- cuit in figure 1-170 except that the bridge stage is operated in push-pull. With different values of inductance, the circuit provides reliable frequency control anywhere in the v-h-f range, from 200 me on down. Probably its most practical application is as a harmonic test oscillator. The upper fre- quency obtainable is not limited by the circuit itself, but by the resistances and shunt capaci- tances of the crystal units. OTHER MODIFICATIONS OF THE CAPACITANCE-BRIDGE OSCILLATOR 1-375. A number of capacitance-bridge modifica- tions have been successfully attempted, four of which are illustrated in figure 1-172. The circuits are largely self-explanatory, and will not be dis- cussed here. Probably of most importance is the electron-coupled circuit, since it permits frequency multiplication in the plate circuit. The triode con- nection of the crystal circuit probably prevents the crystal, itself, from being operated at frequen- cies above 75 me. The Butler Oteilioter 1-376. At the present time, probably the most widely used of the series-mode oscillators is the Butler, cathode-coupled, two-stage oscillator. The basic design and equivalent circuits are shown in figure 1-173. Although the single-tube, trans- former-coupled type of oscillator will probably out- rank the two-tube circuit eventually, the Butler oscillator is the more popular at present because of its simplicity, versatility, frequency stability, and, of most importance, its comparative reliabil- ity. With the older types of crystal units, it was generally found that the Butler circuit was the least critical to design and to adjust for operation of the crystal at a given harmonic. The balanced arrangement of the circuit and the fact that twin triodes can be obtained in a single envelope con- tribute a saving in space and cost, and permit the use of short leads. For greater frequency stability than is normally to be had from parallel-mode oscillations, the cathode-coupled circuit can be used quite satisfactorily at any of the lower fre- quencies provided the resistance of the crystal unit is not greater than a few hundred ohms. However, the power output is small by comparison WADC TR 56-156 244 Figure 1-172. Miscellaneous with that of the Mille'r circuit for the same crystal power, and the broad bandwidth without plate tuning" of the Pierce circuit is not matched. The Butler circuit is usually designed for class-A op- eration, but class C is possible if greater output and plate efficiency are desired. The output may be taken from almost any part of the circuit — the plate or cathode of either tube. Quite often the cathode follower, V,, in figure 1-173, is a pentode, with the screen, control grid, and cathode forming a triode section electron-coupled to a plate circuit that usually is tuned for frequency multiplication, although the electron coupling can be employed simply to obtain greater output amplitude and to isolate the load from the oscillator circuit. At very high frequencies, where the shunt reactance of the crystal unit approaches the magnitude of the series-arm R, the operation is generally improved by shunting the crystal unit with an inductor that Section I CrycHii Oscillators capacitance-bridge oscillators is antiresonant with the shunt capacitance of the crystal at the operating frequency. When properly designed and adjusted, the two tubes operate 180 degrees out of phase into resistive loads, and the crystal unit acts as a pure resistance. 1-377. In figure 1-173, V, is connected as a grounded-plate cathode follower. The V, output current, I,., enters the feed-back path through the crystal unit, which is operating at series reso- nance. The impedance of the crystal unit is thus approximately equal to the equivalent series-arm resistance, Vj, a grounded-grid amplifier connected in the feed-back circuit, is excited by Ij, the com- ponent of I,, that passes through Rj. I,, 2 , the re- maining component of the feed-back current passes through Vj. It can be seen that the inpat voltage of V„ Eg,, is equal to the output voltage of V, if we assume that the coupling capacitance is infinite. The plate circuit of V, is broadly tuned to the de- WADC TR 56-156 245 Section I Crystal Oscillators II O. N WAOC TR 56.156 246 figurm 1-173. (A) Bosk diagram of Burtar two-stage cathode-coupled oseiilalor. (B) Equivalont r-f circuit of Butlor osciffator. Current arrows iadieato instantaneous electron Bow when r-f voltoges have poiarHtos shown. (C) SimpUBod equivalent circuit where gonorator of grounded-grid ampliBor Is replaced by a negative resistance. ID), (E), (f), and (G) Progressive simplHIcations el equivalent Butler tircult sired frequency. If Zp were simply a resistance, the circuit could still oscillate at the first crystal harmonic. If the crystal unit were shorted out, the circuit could also oscillate, but with the frequency controlled by the tuned plate circuit. R, and R, are usually equal, havintf values between. 50 and 200 ohms. V, and V, are also usually of the same tube type. As will be seen, frequency stability is im- proved with largre values of transconductance. Note in figure 1-173 (B) that the r-f plate current in Vi is greater than that in Vj. As in all other vacuum- tube oscillators, there are two fundamental equi- librium conditions to consider: the over-all gain must equal unity, and the over-all phase shift must equal zero. We shall first consider the factors affecting loop-gain. LOOP GAIN OF BUTLER CIRCUIT 1-378. At equilibrium we can say that 1 Gj Ga (respectively) 1—378 (1) The immediate problem is to find the values of Gk in terms of the circuit parameters. First, referring to figure 1-173 (ignore the capacitance C,ei in cir- cuit (B) ), assume that the voltage across R^ is approximately equal to Ep, then E,i = Ep - E, 1-378 (2) Section I Crystal OscHlotort Ip. = Ml t:i 1 Hip Ml + 1 R„ Mi + 1 + Zk Note that with Zk fixed by the external circuit, the plate current is related to the excitation volt- age, Ep, in such a way that the tube behaves as if it had an effective amplification factor of — Ml + 1 R and an effective plate I'esistance equal to — Ml + 1 This resistance is given the symbol Z, in figure 1-173. If an additional resistance, Rl, were con- nected between the plate of Vi and r-f ground, Zi would equal ^ . Now, to find the value of Ml + G, = Eg,/E„ we start with E^ = I* Rj 1—378 (7) El = le R — Ma E*>+ Ip2 Rps + Ipa Zi, 1—378 (8) I. = la + I, p2 1—878 (9) and Ip» _ E ,2 Ma Ei 2 _ (M2 I) Ega Rpa *1" Zl Rpa "I" Zl 1—378 (10) El = Ipi Zk and 1—378 (8) Ip. M. E,i _ M. Ep — M.Ip.Zk Rpi + Zk Rpi + Zk M. Ep Rpi -b Zk (mi + 1) 1—878 (4) On rearranging equation (10) to express E,, as a function of Ip„ and substituting this function for Efi in equation (8) , we have El = IpR + Ip*[- ^ + Zl] 1—378 (11) On combining equations (8) and (4) El = Ml Ep Zk Rpi + Zk (mi + 1) and 1—378 (5) G, Ei/E = ^ — gml_Zk — Rpi H- Zk (M. + 1) 1 + gml Zk 1—378 (6) The approximation in equation (6) is made on the assumption that (/n -|- 1) ~ /*,. If the numerator and denominator of equation (4) are divided by (/*, 4- 1), we have From equation (11) we find that the equivalent generator of V, can be represented by an equiva- lent negative resistance ~ Ma (Rpa + Z;,) Ma + 1 1—378 (12) It can be seen that p is smaller in magnitude than (Rp, -f- Zl) , BO that the total Vi branch resistance is positive. Defining the V> branch impedance to be Z„ we have Z, = p -j- Rp2 4- Zi, = 1-^78 (13) Ma + 1 WADC TR 56-156 247 Section I Crycto! OKillators On substituting equations (7) and (13) into equation (10), we have Ip2 = 1-378 (14) 1^2 and Q /> _ El Ml Zit Egi Rpi + Zk it will be found that 1—378 (23) so that equation (9) may be written Gi' G." Gg G* = G, G, G, = = ( Zj + R2 \ r Ri Zl y )l. 1—378 ( 15 ) [.(Z2 4 " R2) R "b R2 Z2 Ml Zk Rpi + Zk On substituting equations (13), (14), and (15) into ( 11 ), _ I2 R (Z2 + R2) , I2R2Z2 1-878(16) 1—378 (24) Since Rpi and R,Zi, are very large compared with Zk and [(Z, + R,) R + Rs Z,], respectively, we can simplify equations (21) and (24) by writing R2 Zl Zk gml (Z2 + R2) R + R2 Z2 1—378 (26) Thus G2 Eg2 R2 Z2 El R (Z2 ”i" R2) "b R2 Z2 1—378 (17) To find G3 (= Ep/Eg2), we see that On dividing both numerator and denominator by (Z, 4- R,) and substituting for the values of Zgj and Zf as defined in equations (6) and (7) of fig- ure 1-173, equation (25) can be simplified some- what. Thus Ep = Ip2 Zl 1—378 (18) which, on substituting the value of Ip2 given in equation (14), becomes Ep = 1—378 (19) Dividing by Eg2 (= I2 R2). we have G3 = 1^ = 1-378 (20) J!,g2 02 The conditions for equilibrium as expressed by equation (1) are thus found to be Gi G2 G3 — Ml Zk R 2 Zl _ - [Rpi -h Zk (mi + 1)1 [(Z2 4- R2) R 4- R2 Z2] 1—378 (21) By a slightly different approach, in which the equilibrium is expressed as Gi' Gi" G2 G3 where Egi El Eg2 _ Ep Egi El Eg2 1 Zg2 Zl Zk gml ^ j 1—378 (26) Z2Z, 1-379. The design of a Butler oscillator must be such that under no-signal conditions the left side of equation 1 — 378(26) is greater than unity. As oscillations build up, the principal effects will be a decrease in the effective gm, and go,, as the sig- nal swings farther into the lower bend of the EcR curve. How large the equilibrium amplitude will be depends upon how much greater than unity the left side of equation 1 — 378(26) is at the Start. The larger the left-side magnitude, the greater must be the decrease in g,„,, and hence the greater the equilibrium activity must be. If the oscillator is to operate class A, as is usual, the gain equi- librium should very nearly hold for no-signal con- ditions, with due allowance made for a maximum Z, (= R 4- Zgj) when R = R„, the maximum series resistance permissible for the particular type of crystal unit chosen. With all else constant, maxi- mum activity is to be obtained when g^, and gm, are maximum under no-signal conditions. Assur- ance that the crystal unit will not be driven beyond its rated power can be approximately predicted from the plate characteristics of the tube to be used. If no grid current is drawn, the bias on Vi will be R2 Zl — (Z2 R2) R — R3 Z2 R2 Zl 1—378 (22) Ep, = - Ib, Ri 1-379 a) where Ipi is the average d-c plate current of WADC TR 56-156 248 Vi. Grid current can be drawn if [(niax)Ep — (max)E,] is greater than IbiRi, in which case the bias will be Eoi = (max) El — (max) Ep 1 — 379 (2) Greater amplitude stability is achieved if R, is sufficiently small for equation (2) to apply. Using the appropriate equations in paragraph 1-378, a maximum value of E, can be determined that will not allo w the crystal current, Ic, to become greater than V^cm/R. where Pen. is the maximum recom- mended power level of the crystal, and R is any crystal resistance between Rm and Rm/9. Very pos- sibly, a twin triode may be preferred, or perhaps the choice of tubes will be dictated by the h-f type of tube most readily available. Rj and Ra are to be kept as small as possible in the interest of fre- quency stability. In the final analysis, the plate voltage permitting an optimum output for the average crystal unit, without risking an overdrive for any expected value of crystal R, is most easily checked by experiment. DESIGN CONSIDERATIONS TO MAXIMIZE THE FREQUENCY STABILITY OF THE BUTLER OSCILLATOR 1-380. If the cathode follower operates into a purely resistive network, as is indicated in the equivalent circuit of figure 1-173 (B), maximum stability in the phase characteristics is obtained. As nearly a resistive circuit as possible is desir- able, for under these conditions the frequency is independent of the plate resistance of the tubes, and a small increment of reactance requires the least adjustment of the crystal to restore a phase equilibrium. E., the output voltage of the cathode follower, is in phase with the excitation voltage, Ep. In a resistive circuit, Ep will be 180 degrees out of phase with E, 2 . Thus, Ep. must be 180 de- grees out of phase with E,. Since Egj is the voltage of ground with reference to the cathode of Va, and El is the voltage of the cathode of Vj with refer- ence to ground, U and U must be in phase. For example, imagine that R is zero, then Ri and Ra could be assumed to be two halves of a single re- sistance. In this case Ega would equal — E,, and the proper phase relation would exist. 1-381. To maintain as nearly as possible a resis- tive circuit, Zp must tune as broadly as is practi- cable : the tendency of the input capacitances of V, and Va to shift the phase must be compensated ; the transit time of the vacuum tubes must be mini- mized ; and, if an inductor is connected across the crystal unit to antiresonate with the shunt capaci- WADC TR 56-156 Section I Crystal Oscillators tance, C„, the resulting parallel-resonant circuit must also tune very broadly. The tuned plate cir- cuit, Zp, must be sufficiently selective to ensure that the circuit can oscillate only at the desired harmonic of the crystal frequency, but, beyond this, any increase in the tank selectivity only re- sults in a greater phase shift, and consequently a greater frequency shift, for a given percentage change in the plate capacitance. The use of a damping resistance as indicated in figure 1-173 (A) , or a low-Q coil, will broaden the tuning of the tank. The stray capacitance from the plate of Vj to ground should be kept to a minimum. 1-382. To annul the input capacitance of Vj, which is equal to the total capacitance between the cath- ode of Vj and ground, we can connect an inductor in series with Rj, or replace Rj with a low-Q in- ductor and employ gridleak bias for Vj (while keeping the grid at r-f ground by the use of an r-f bypass capacitor), or shunt Rj with an inductor in series with an r-f bypass capacitor. In any event, the inductor is to be antiresonant with the cathode- to-ground capacitance at the operating frequency. With Ra acting as a damping resistance, a broad- band response is ensured for the antiresonant combination. 1-383. The grid-to-cathode capacitance and the cathode-to-ground capacitance of V, can also be annulled by the use of antiresonant inductors. However, a more effective and economical method is to design the circuit so that the two cathode capacitances of V. neutralize each other regardless of the particular frequency. The grid-to-cathode capacitance, Cgd, is illustrated by the dotted-line circuit in figure 1-173 (B). The voltage across Cgci is Eg., so that the leading component of current through the grid circuit is For convenience, let it be imagined that all of Igg flows through R and Rpj in completing its circuit. If it is not to upset the phases of the voltages across these resistances, Igx must be annulled by an equal lagging current through the R-Rpj-Zp circuit. Thus, assuming the transit-time effect is negligible, the plate tank must be slightly induc- tive if Vj is to operate into a purely resistive load. This much can be controlled by the adjustment of the Vj plate circuit. With the circuit properly ad- justed, it can now be imagined that Igj is no longer a part of !„ but circulates directly through Zl and Cp,i in series. The design problem is to ensure that no part of I,, flows through Ri or V„ but returns 249 Section I Crystal Oscillators to Zl by flowing entirely through the Vi cathode- to-ground capacitance, C, — not shown in figure 1-173. Furthermore, the design should be such that 1,1 is all the current that flows through Ci ; other- wise, there will be a net unneutralized leading com- ponent upsetting the voltage phases in the rest of the circuit. With proper neutralization, the only reactive current will be confined to a series circuit comprised of Cgci, C„ ground, and an effective in- ductance shunting Zj.. The voltage across each of the reactive impedances due to I,, will be equal in magnitude and phase to the voltages caused by the in-phase currents flowing through the correspond- ing resistive impedances. Thus, to neutralize the circuit, the leading current, I,,, flowing through C, must of itself produce the voltage E,. This occurs when El — Ig, Xci 1 — 383 (2) or, using equation (1) to replace I,,, El = 1_383 (3) Xogl Now, El = = gnu Eg, Z, 1-283 (4) Using equation (4) to eliminate E, in equation (3), we have 1, E Z Xci gml ^*1 Ok — = or g™. Zk = ^ 1-383 (5) '-'I Equation (5) defines the ratio for the V, input to output capacitance that should exist for maxi- mum frequency stability. 1-384. To minimize the tendency of the transit time to cause the respective plate currents of V, and V, to lag the equivalent generator voltages, small-dimensioned h-f tubes should be used, and the plate voltages should be as high as practicable. If the additional expenditure in the design and production of the circuit are warranted, suitable networks can be devised to neutralize the transit effects. 1-385. At the higher frequencies the series resist- ance of the crystal unit tends to increase, since the lagging component of current through the series arm must increase in order to annul the increased leading component through the shunt capacitance, C,. For this to occur, the frequency may need to be increased considerably above the natural resonance of the motional arm, so that the effective series resistance approaches the value of a parallel-resonant impedance. To reduce the re- sistance to the series-arm value, €„ should be annulled by an antiresonant inductor having an inductance L„ = 1-385 (1) w Co To prevent the crystal shunt reactances from be- ing more frequency sensitive than the plate circuit of V 2 , a shunt resistance should also be connected across the crystal unit to dampen the LoC, tank. A suitable resistance, Ro, that can interfere very little with the crystal stabilizing effect is R. = 6 R™ 1—386 (2) where Ro. is the rated maximum permissible crys- tal series resistance. STABILIZING EFFECT OF CRYSTAL IN BUTLER CIRCUIT 1-386. From equation 1 — 241(2) we found that for a crystal operating at series resonance the fractional change in frequency required to produce a small change in phase, is expressed by dtp _ Ro (i> dfl 2-\/L/C where Ro is the total resistance the crystal faces, including the crystal’s own resistance, and L and C are the series-arm parameters of the crystal unit. In the Butler circuit (refer to figure 1-173) the crystal operates into a resistance R. = Z, + 1-386 (1) where Z, is the resistance of the feed-back circuit, R Z and p V. is the output resistance of Vi. On Ki "I substituting the values for Z, and Zt, we have R. - R + Rj (ais + 1) + Rps + Zl + Ri Rpi 1—386 (2) Ri (mi + 1) + Rpi If we assume that Rp, = Rp 2 = Rp > > Zl, and that WAOC TR 56-156 250 Pi = Ma = M > > 1» equation (2) becomes R, = R 4- RaP + Rp RiP "I" Rp or Ro = R + Ra + Ri Rag. + 1 Rig. + 1 1—386 (3) From equation (3) it is seen that the Q of the crystal circuit, and hence the frequency stability, is to be improved if Rj and R, are kept as small as possible and the transconductance of each tube is high. If R, and R, are of such values that the denominators in equation (3) are large compared with 1, a limiting value is approached, where (max) R„ = R -f — 1—386 (4) g. DESIGN PROCEDURE FOR BUTLER OSCILLATOR 1-387. In considering the use of a Butler oscillator for controlling frequencies below 20 me, the prin- cipal factor to consider is whether the frequency stability required is greater than that which is normally obtained with a Pierce circuit. If not, there is little to gain by using two tubes and a frequency-sensitive tuned circuit, unless it is very important that the waveform in the output be more nearly sinusoidal and less influenced by the variations in the crystal resistance. The frequency stability of an average Butler, circuit can be ex- pected to be approximately 0.0005 per cent as com- pared with a stability of approximately 0.001 to 0.0015 per cent for an average Pierce circuit. Above 20 me, the principal competitor of the But- ler is the transformer-coupled type of oscillator. The chief advantage of the Butler is its relative ease of adjustment and dependability. A border- line replacement crystal unit or an aging crys- tal unit, as a general rule, is more likely to be operative in the two-stage, cathode-coupled circuit than in any of the other types of v-h-f oscillators. Once the Butler circuit has been selected as the most appropriate to use, a crystal unit that has been series-tested at the intended frequency should be selected. The required minimum frequency tol- erance and the operating conditions to be expected determine whether the crystal unit, or perhaps the entire oscillator, is to be oven-controlled. For the next step, it is probably best to select the types of vacuum tubes to use. Insofar as space, weight. Section I Crystal OKillaters and cost are concerned, a single tube envelope for both amplifler stages is desirable. On the other hand, it may be found that a more balanced ar- rangement and more direct circuit connections can be had with separate tubes, particularly if the crystal unit is to be oven-mounted. The transcon- ductance and plate resistance of the tubes should both be high, for maximum frequency stability. For the same amplification factor, the tube with the larger g„, is usually to be preferred. For h-f and v-h-f operation miniature tubes are prefer- able, in order to reduce the transit time and the electrode-to-ground capacitances. For class-A op- eration, both tubes can be of the same type. For class-C operation, the power rating of the cathode follower should be greater than that of the grounded-grid amplifier. For maximum stability it may be desirable to isolate the load from the rest of the circuit, or to tune the load circuit for frequency multiplication. In this case a pentode can be used for either the cathode follower (usu- ally) or for the grounded-grid amplifier, with the load taken from the electron-coupled plate circuit, and with the screen grid serving as the oscillator plate. Either pentodes or triodes can be used in the basic Butler circuit, as desired. In the v-h-f range, triodes have the advantage of smaller tran- sit-time effects. Assume that it is intended to oper- ate the tubes class A. To reduce the transit time, to increase g„„ and to permit a minimum value of Zl (low-Q tank), the plate voltages should be as high as practicable. Determine the values of R, and R, that will provide a normal cathode bias for class-A operation. With all else equal, the feed- back transmission losses are a minimum if Ri = Rj. For class-C operation, Rj should equal approxi- mately 4Ri. Assume that R of the crystal unit is the maximum permissible value, and that the ef- fective gni’s of the vacuum tubes are 25 per cent less than their rated values for class-A operation at the selected plate and grid voltages. With these assumptions, determine the value of Zl that is re- quired to make the gain equation, 1 — 378 (26), hold. Rg should be large compared with Zp, so that Zl Zp. The plate tank represented by Zp can be designed as a high-Q circuit, antiresonant at the operating frequency, and shunted by a simulated load resistance, Rl, much smaller than the anti- resonant impedance of the tank, itself. In this case, Zl Zp =« Rl. The approximations above will be sufficient to build an experimental circuit that should oscillate in a free-running state. A variable resistance can be connected to simulate a crystal unit at series resonance. By varying the simulated crystal resistance over the range possible for a ran- WADC TR 56-156 251 section i Cryctai Oidllatert dom selection of crystal units, Ri, in the plate cir- cuit can be adjusted, if necessary, to ensure ttiat oscillations occur at ail possible values of crystal resistance without driving the crystal at a higher than recommended level. The empirical optimum value of Ri. can be accepted as the value of Zl to achieve in the design of the output circuit of the grounded-grid amplifier. The actual design of the output stage depends, of course, upon the type of load into which the oscillator is to operate. The im- portant consideration is that an effective resistance having the value of the experimental Rl is to be introduced in one way or another across the plate tank. The final problem is to neutralize the various circuit capacitances. In neutralizing the V, cathode capacitances, the adjustment which permits the feed-back circuit to be purely resistive can be ex- pected to coincide with the conditions for maxi- mum output amplitude and maximum crystal cur- rent. The design procedure discussed above should be accepted simply as a suggestion. Individual engineers may well prefer that primary attention be given to fitting the design to meet special requirements. MODIFICATIONS OF THE BUTLER OSCILLATOR 1-388. As in the case of other conventional oscil- lator designs, the number of modifications of the Butler circuit appear to be unlimited. In figure 1-174, the basic electron-coupled circuit is shown in (A), and a circuit employing a common ground return for the two tubes is shown in (B). In the electron-coupled circuit, the load, represented by Rl', is effectively isolated from the oscillator cir- cuit, in which the screen of Vi serves as the cathode-follower anode. The plate circuit of V, can be tuned to the second or third harmonic of the oscillator frequency, if desired, in which case Vi should be operated class C. For maximum output voltage, the plate impedance (rf Vi should be high. In figure 1-174 (B), the low-Q inductor, Li, is flgurm 1-174. (A) ffectroii-coup/ed Buffer clniHt. (B) Buthr drcuH having common cathodn ground roturn WADC TR 56-156 252 antiresonant with the distributed capacitance, C„ as is L. with Ce. Gridleak bias is employed with both tubes. However, the grid of V, is still kept at r-f ground through the bypass capacitance. It can be seen that equation 1 — ^78 (26), Ztf Zt, Zi, gmi ^ . Z,z, when applied to the common-ground return cir- cuit, becomes = 1 1—388 (1) Zj since Z* is equal to Z,. Also, since Z, = «« — , since Z^ where Rj now represents gm> tit the antiresonant impedance of the L,C, combina- tion, it can be shown that equation (1) can be expressed as Zl g„.i - ^ 1 1-388 (2) Ra g,ia If the product Rigms is large compared with 1, at equilibrium gmi must approximately equal 1/Zi,. 1-389. The schematics of a number, of Butler cir- cuits employed by the military services are shown in figure 1-176. Circuit (A) is a receiver hetero- dyne oscillator that can be switched from crystal to manual operation simply by shorting out the crystal. C, is an r-f bypass capacitor that prevents the inductor L, from shorting out the cathode bias developed across Ri. L, is designed to be antireso- nant with the grounded-grid amplifier cathode-to- ground capacitance. C, is inserted to resonate with the distributed inductance of the crystal leads and feed-back circuit. Cj is a split-stator capacitor which permits the use of a grounded rotor, tiiereby reducing intersectional capacitances. Since the r-f current through C, is small, wiping contacts can be used for grounding without introducing noise. C, is simply a trimmer which permits adjustment of the effective inductance of L,. The plate side of C, effectively has a relatively large fixed compo- nent due to the stray plate-to-ground capacitance. To keep the plate tank balanced, an equal amount of fixed capacitance, Cg, is added to the other side of Cg. Rg is added to suppress parasitic oscillations. As can be seen, the output is obtained by split- load operation of the cathode follower. Since the output is delivered to a mixer circuit, where the effective load might be expected to undergo slight Sacrien I Crystal OsdNotors changes, it is probable that the loaded cathode- follower plate circuit is less frequency sensitive than the finely balanced tank in the V, plate cir- cuit. Since the balanced tank must be sufficiently selective to stabilize the frequency during manual operation, it cannot be loaded as would normally be done. Although the crystal is not active in the circuit during manual operation, it cannot be removed without resulting in an increase in fre- quency. This is due to the decrease in the cathode- to-ground capacitance that results when the crystal unit is removed. 1-390. Figure 1-175 (B) (C) (D) (E) (F) is a composite arrangement of five different oscilla- tors, none of which have all the components shown. For example, the B+ return of V, is through R. in (B), (E), and possibly (F) ; it is through R, in (C), and also in (D), except that R, and R, are one and the same in the latter circuit, although the actual circuit is not indicated in the schematic shown. In circuit (E) the F, output is cathode- coupled to a mixer tube (6AK5W). An r-f choke is connected between the cathode and ground, not the resistance R,o. In circuit (B), Cg is actually composed of two 1.5-/v»f capacitors in series. The F, output is developed across a tuned tank identi- cal with and also inductively coupled to, the plate tank in the V, plate circuit. L, in circuit (B) thus serves as a transformer primary. The Fg output is fed to the grid of one and to the cathode of a second 6AG5 mixer stage. The heterodyned out- put of the first is 20 to 30 me, and that of the second is 4.8 to 5.7 me. 1-391. The circuit shown in figure 1-175 (G) is a carefully designed experimental model that was built and tested during an investigation of h-f and v-h-f oscillators by a research team headed by W. A. Edson at the Georgia Institute of Technol- ogy. A breadboard model of this circuit was oper- ative at frequencies as high as 150 me, with crystal resistances as high as 500 ohms. Circuit (G) was found to have a frequency stability of 0.22 parts per million per volt change in the high-voltage sup- ply. The frequency was controlled at 126 me, the ninth harmonic of a 14-mc fundamental. The shunt capacitance of the test crystal was 12 /i/if, and the series resistance after tuning out the capacitance was 300 ohms. Oscillations could not be sustained at plate voltages below 60 volts, and the frequency instability increased greatly at voltages above 85 volts. Note that circuit (K) in figure 1-175, which has been designed to operate with low- and me- dium-frequency, fundamental-mode crystal units, is not a true Butler circuit in that neither tube is operated as a cathode follower. WADC TR 56-156 253 Section I Crystal Oscillators F» or figure 1-175. ModWeatioas of Butler oscillator. Dotted lines In circuit (G) indicate stray capacitances Fig. Equipment Purpose Fi Fj F, Ft CR Ri Rs Ri Rs Rs Rs Rr Re (A) Radio Receiver R-266/ URR-18 Hetero- dyne oscillator 20.3- 34.9 2Fi ■ ■ CR-24/U (5th mode) 0.18 0.01 6.8 0.18 12 6.8 ■ 1 (B) Receiver Trans- mitter RT-178/ ARC-27 3rd trans- mitter osc or 2nd re- ceiver osc for hetero- dyning 25.7- 34 7 F, T NA CR-23/U 1 1 100 1 0.16 NA 12 NA (C) Receiver- Trans- mitter RT-178/ ARC-27 1st guard receiver local osc 37.266 NA 2Fi CR-2S/U 0.39 0.12 47 0.22 0 12 6.8 1 (D) Receiver- Trans- mitter RT-173/ ARC-33 1st mon- itor osc of trans- mitter M.O. 0.8333 F, 4F,- 18Fi Fi CR-28/U 8.8 S3 100 1 1 Same re- sistor as Rs 1 1 ^^ Receiver- Trans- mitter RT-178/ ARC-83 Guard- channel hetero- dyne oscill- ator- doubler 65 . 668- 58.169 F. NA 2Fx and 4F, CR-32/U 0.12 8 2 38 1 0.33 NA 27 00 (F) Radio Receiver R-252A/ ARN-14 Ist heter- odyne oscillator 44.276- 57.276 F, NA 2F, CR-23/U ■ 8.2 ■ 0.1 1 7 00 ■ Circuit Data for Figure 1-176. F in me. R in kilohma. C in /i/il. L in ah- NA (not applicable) means that no connections of any kind exist between points indicated. Question mark (7) indicates that schematic of the associated part of the circuit is not available. WAOC TR 56-156 254 Section I Crystal Oscillators cs figurm I- 175. Contlnu 0 d R« Rio o C2 Ca Cl Cs lO Cr Cs C, L, u La Li La V, Vj ■ 200 14.3- 43.6 0.5- 3.0 15 1000 1000 ■ 1 1.2 ■ ■ ■ ■ GL6670 1 NA 1-8 1 1500 47 0 75 NA NA NA NA 00 00 NA 1 12AT7 NA 27 0 20 3000 100 NA 24 500 20 0 00 00 00 ■ ■ 12AT7 NA 470 0 104 ■ 470 470 100 25. 000 170 0 00 CO 00 ■ 1 6670 8.2 Cath- ode of mixer 1 5 1 10 NA 1.6- 5 00 10 10 00 CD 1 1 1 5670 ? ■ i 12 2000 100 NA ♦ ? ? ? ■ 12AT7 WAOC TR 56-156 355 ^2,000 ^jpo Fig. Equipment (G) Experi- mental osc (H) Radio Receiver R-540/ ARN-14C (I) Radio Set AN /ARC- 34 (XA-1) (J) Radio Set AN/ARN- 21(XN-2) 44.275- 57.276 (14 crystals) 55.67- 2F, 68 . 17 and 4F, CR R. R. R. 9th har- monic; 0 05 0 05 5 series resist- ance equal to Rs CR-23/U 100 8.2 8.2 CR-32/U 0 22 0.66 33 CR-23/U 0.22 ■ 10 R, R, osc and multiplier Circuit Data for Figure 1-175. F ir me. R in kilohma. C in «*(• L in iih. NA (not appiicable) means that no connections of any kind exist between points indicated. Question mark (?) indicates that schematic of the associated part of the circuit is not available. WADC TR 56-156 WAOC TR 56-156 257 Section I Crystal Oscillators Fig. Equipment Purpose a D D a CR R, Q R, R. R» R. R, R. (K) Signal Generator SG- 34(XA)/ UP Lf-F and m-f osc 0.10 0 18 1 75 1 85 1.90 1.95 1 1 1 CR-16/U (LF.) CR-19/U (HP,) 10 15 0.027 560 39 1000 0.27 1 (L) Signal Generator SG-13/ ARN 200-mc generator for mixing with lower freq. signals 60 4Fi 1 1 CR-23/U 58 3.3 0.015 0.01 0.27 0.01 0.27 1 Circuit Data for Figure 1 — 175. F in me. R in kilohma. C in L in ah- NA (not applicable) means that no connections of any kind exist between points indicated. Question mark (?) indicates that schematic of the associated part of the circuit is not available. SAKSW Ffffvre 1-175. ConUnumd (L) Transformer-Coupled Oscillator 1-392. At the present time, the transformer- coupled crystal oscillator (see figfure 1-176) is not being widely used. It was during the v-h-f oscilla- tor investigation at the Georgia Institute of Tech- nology for the Signal Corps in 1950, mentioned in the last paragraph, that the transformer-coupled oscillator appeared to be the most promising for all-around versatility and general-purpose use. First, there is the advantage of a single-tube oscil- lator. Secondly, for low-power output (about four times the crystal power), the frequency stability has been found to be slightly superior to that of the average Butler circuit. Thirdly, with properly desigfned phase-compensating networks, an un- tuned pass band of 10 me is possible. Finally, with a relatively small sacrifice in frequency stability, the circuit design can be such that the power out- put is increased several fold without exceeding the recommended maximum drive level of the crystal unit. Although the transformer-coupled oscillator can perform satisfactorily at lower frequencies, its chief application is for control and generation of harmonic-mode frequencies above 20 me. The oscillator is generally designed for class-C opera- tion. A significant disadvantage is that the circuit' design for optimum performance characteristics — characteristics that can be approximately dupli- cated from one oscillator to another of similar de- sign — is generally more difficult to achieve than in other oscillator circuits. This is due chiefly to the difficulty in predicting the effective input im- pedance of tubes operated class C at frequencies where transit-time and stray-capacitance effects become appreciable. As a result, the theoretical and actual equilibrium conditions frequently are found to differ to a greater degree than in the average series-mode oscillator. More cut-and-try experimentation may prove necessary than would otherwise be the case. The operating principle of the grounded-cathode, transformer-coupled oscil- lator is closely allied to that of the grounded-grid and grounded-plate versions. The discussion and equations for the transformer-coupled oscillators WADC TR 56-156 258 Section I Crytfal OscillcMon R. Rio D D a B B B B B B D B B B Li V, V, 1 1200 10,000 1 1 1 1 1 1 1 1 1 1 1 1 12AU7 1 1 500 1.7-8. 7 500 100 12 500 1 1 1 1 1 1 1 1 12AT7 are, for the most part, based upon the analysis by W. A. Edson. PHASE CONSIDERATIONS OF TRANSFORMER-COUPLED OSCILLATOR 1-393. Referring to figure 1-176, the useful load, represented by Rl, is connected across the second- ary of the plate transformer. The chief function of R, is to improve the frequency stability by low- ering the resistance of the crystal circuit, and to improve the amplitude stability by reducing the effect of variations in the input resistance of the tube. The parameter a is simply the constant of proportionality relating Ri. to R,. C, and C, are capacitors for tuning out the leakage inductance of the plate and grid transformers, respectively. The leakage inductance is equal to the high-side inductance multiplied by (1 — k’), where k is the coefficient of coupling. It can be directly measured at the low side of the transformer when the high side is shorted. Both transformers can be, simply, tapped coils. The crystal impedance is assumed to be the series-resonance impedance, R. The effec- tive turns ratios, N,, and N„ can be so chosen that Cp, and C„ annul each other’s effects. With the circuit properly designed, the tube operates into a resistive load, C,, being antiresonant with the dampened coil L„. I., I,., E,„ and E. (the voltage across the crystal) are in phase with E,,. The grid transformer thus provides the required 180-de- gree phase shift between E,, and E,. 1-394. Where the resistance of the crystal series arm is not small compared with the shunt react- ance, Xco, the effects of C„ can be annulled by the conventional method of connecting an inductor across the crystal unit, by means of mutual induct- ance between the plate and grid transformers (to be discussed in connection with the grounded-grid oscillator), or by balancing the effects of C, against those of C„t. When the circuit is properly balanced, the crystal unit operates at the resonant frequency of the series arm. When C„ is balanced against Cp^, the leading component, I„, of the cur- rent through the crystal — that part through C„ — passes in its entirety, through Cj and the primary of the grid transformer. Similarly, the current through C,„ Ipp, passes in its entirety, through the secondary of the grid transformer. For this to occur, the voltages induced by the two currents in each section of the grid transformer must exactly figun 1-176. (B) Equivo/ant paralM- rasenant picrta drcuH wh»n crystal capac- itanea, C„ Is balancad by plats-to-grid FIgurm 1-176. (AJ Basic circuit of transformor-couplod oscillator capacitanco, WADC TR 56-156 259 Section I Cryctol Occillaion annul each other. The transformer then appears as a short circuit to both currents. As indicated in fig- ure 1-176 (B), C„ becomes equivalent to a capaci- tor shunting the ground-connected half of Lp, thereby effectively increasing L,,, and C,„ becomes equivalent to an increase in C,, equal to Cp*. The balanced state is reached when I = N I ■^O *pg or wCqEl — NgtoCpg Ep or C PC Co ^ Ng ■ Ep Co NgNp 1—394 (1) 1—394 (2) 1—394 (3) Equation (3) can generally be realized with prac- tical values of Ng and Np if Cpg is on the order of 0.05 to 0.1 /ifif. Such values of Cp, can be ob- tained with screen-grid tubes but not with triodes unless a d-c blocked inductive arm is connected between plate and grid to annul most of the capacitance. (eff)R.E,^ Eg* (eff)Rg Ng* 1—395 (6) where (eff) Rg is the effective grid resistance which takes into account the gridleak losses and the transit-time loading. For small values of Ng, the Rg' losses can be considered negligible. The plate power can be assumed to be. Pp = I. E. = El^ (Rl -1- R ■+• Ri) Rl (R -|- Rl) El* (R -|- Rl -t- a R,) a Rl (R Rl) and V ^ a Np* R, (R -h Rl) ” Pp R H- R( -t- a R, 1—395 (7) 1-395 (8) on combining equations (2), (3), (4), (5), and (8), we find that at equilibrium Gi G2 G3 G4 gp, a Np Ng Rl* R ■+• Rl (1 + a) 1 GAIN REQUIREMENTS OF TRANSFORMER- COUPLED OSCILLATOR 1-395. Referring to figure 1-176 (A), it can be seen that at equilibrium. Gi G2 G3 G4 = ^ where El El ^ _ 2^ Ep El El 1—395 (1) mZp Rp + Zp 3m Zp 1-395 (2) G2 = ^ = — 1—395 (3) Ep Np R. R -j- Rl 1—396 (4) G4 = Ng 1—395 (5) Rg' in equation (4) is the effective resistance of the grid transformer as it appears in parallel with R,. It is equal to E,* divided by the grid losses. Thus, 1—396 (9) Q DEGRADATION IN TRANSFORMER- COUPLED OSCILLATOR 1-396. In Edson’s analyses of series-mode oscilla- tors, he employs a useful term which he calls the Q degradation of the crystal unit. It is defined D = ^ 1—396 (1) Iv where Rg is the total resistance which the crystal must operate into. In the transformer-coupled os- cillator, assuming that the transformer imped- ances that the crystal faces are large compared with Rl and R, of figure 1-176, D R -h Rl (1 a) R 1—896 (2) As discussed in paragraph 1-241, the frequency stability of the series-resonant crystal unit is di- rectly proportional to the Q of the crystal circuit. Q It is therefore directly proportional to -g, where Q is the Q of the crystal, itself. Thus, with a given crystal unit, the frequency stability varies in- versely with D. If the minimum Q of a crystal unit is estimated from the maximum permissible C„, from the frequency, the harmonic, the particular crystal element, and the maximum permissible WADC TR 56-156 260 series resistance, and if the required frequency stability is known, then the maximum permissible D can be determined from the random phase shifts to be expected during: operation. In the averagre cir- cuit, it is sufficient, simply, to keep D as low as possible consistent with the output desired. When R, is expressed as a function of a, R, and D, the loop grain as defined by equation 1 — 395 (9) be- comes G, Gj G3 G4 g„ a Np N, R (D - 1)^ D (a -f- D* ’ Section I Crystal Oscillators Nk be as small as possible. For a griven value of the product N„N, can be a minimum when -, f -v, (see equation 1 — 396 (3) ) is a maximum. The maximum occurs when a = 1. With a, D, and g„, decided upon, equations (2) and (3) give mutually minimum values when N,, = N,. Thus, for broad- band operation, let Zp = Z, D + 1 gn. (D - 1) 1—398 (4) 1^96 (3) LOAD-TO-CRYSTAL POWER RATIO OF TRANSFORMER-COUPLED OSCILLATOR 1-397. The ratio of the load to the crystal power is p /p - ^1- _ (D + a)^ “ I,* R a (a + 1) (D - 1) 1—397 (1) It is generally desired to have the power ratio as high as is consistent with satisfactory frequency stability. With a given value of D, the ratio be- comes large as a is made small. If the design is based on obtaining a griven minimum output with a given value of D when the crystal-unit resistance is a maximum (minimum D), the value of a can be determined by equation (1), and equation 1 — 396 (3) can be used to determine the value of NpN, most likely to produce the required gm for the crystal to be driven at the desired level. BROAD-BAND CONSIDERATIONS IN THE TRANSFORMER-COUPLED OSCILLATOR 1-398. For broad-band untuned operation it is im- portant to have Zp and Z, (the impedance faced by the grid) and the plate and grid capacitances as small as possible. Assuming that R„ is large com- pared with the resistance appearing across the secondary of the grid transformer. N,* R, (R -b a R) a Ri -f- Ri R or N, (D -H 1/a) ‘ g„Np(D-l) Also, Zp can be expressed as 1—398 (1) 1—398 (2) Np (D + a) " gm N, (D - 1) 1—398 (3) To keep Zp and Z( low, it is desirable that Np and WADC TR 56-1 5« 261 D should be as large as possible consistent with the required frequency stability. FREQUENCY STABILITY OF THE TRANSFORMER-COUPLED OSCILLATOR 1-399. The frequency-stability equations for the transformer-coupled oscillator are and 1-399 (2) where Q is the Q of the crystal. When equations 1 — 398 (2) and (3) are multiplied by D, it can be shown the DZp is a minimum when " D = 1-1- VTTa 1—399 (3) and DZp is a minimum when D = l-h V 1 + 1/a 1—399 (4) For both to be a minimum simultaneously, a must be equal to 1, which means that D = 1+ VT = 2.414 1—399 (5) Under these conditions, the power ratio, as given by equation 1 — 397 (1), becomes ^ = 4.12 1—399 (6) * C For larger power outputs, the value of a must be decreased and Np increased. Since the expected variations in the grid capacitance are generally larger than those in the plate capacitance, it is usually desirable to favor the grid circuit insofar as the frequency stability is concerned. 1-400. The grreatest probability that the effect of a random vari4gtion in the grid capacitance will be Section I Cryttol Oecillator* canceled by a random variation in the plate capaci- tance occurs when the fractional change in fre- quency due to the average aC„ is equal to that due to the average aC,. If the average aQ is on the order of 10 times the average aC,„ then Zp should be equal to lOZg. Equations 1 — 399 (1) and (2) will then represent equal average variations in frequency. DESIGN PROCEDURE FOR TRANSFORMER- COUPLED OSCILLATOR 1-401. The procedure to follow in designing a transformer-coupled oscillator depends upon the principal objectives to be sought in the design. That is, some fixed requirement serves as a start- ing point, and the design proceeds from there. One limitation that will be common to all the circuits is that the crystal power rating not be exceeded. This requirement, then, in the general case, can be the initial design consideration. The crystal power is P„ = R = 1-401 (1) Mg Ki Also p _ e,;r ' ' (R, + R)^ Np“ (Ri + R)^ _ g,p^E/Zp^R Np^ (R + R.)^ or Pc E,^ a' R,^' Np“ R D" 1—401 (2) Multiplying equation (1) by equation (2) and tak- ing the square root, we have e E ^ a N P„ = " 1-401 (3) D N,, Now, (gniE/) is assumed equal to (I, .Eg), which, in turn, is principally a function of the excitation voltage and the plate characteristics of the tube to be used. By equation 1 — 395 (9) (also by equating equation (1) to equation (2) ) 3m Rc a R,“ Np Ng 1-401 (4) Rc, remember, is equal to RD, the total resistance that the crystal unit operates into. From equation (4) it can be seen that if Rc(= R -f Ri(l + a) ) is large compared with the maximum R(= Rm), the equilibrium transconductance will be approxi- mately the sanqe for all values of R. On the other hand, if the minimum D is small, the value of gm at minimum R may be as much as one-half its value for R = Rm. Note that some change must occur in gm, and hence in Eg, if R varies and the rest of the parameters remain constant. To ensure class-C operation for all values of R, let the class-A value for g.„ equal twice the equilibrium g„i according to equation (4), with R,. assumed to be a maximum. With the circuit so designed, the amplitude of the oscillations will build up until the tube is cut off a fair proportion of each cycle. Even if class-A oper- ation is desired, a reasonable difference should be allowed between the rated transconductance of the tube and the estimated equilibrium value when R,. is maximum. This should be sufficient to allow for all expected tolerances in the plate characteristics and in the tuning of the oscillator circuit. The per- centage variations in R from one crystal unit to the next is not quite as great in the v-h-f crystals as in the lower-frequency elements, since it is more important that the maximum permissible resist- ance be kept as small as practicable. 1-402. The ideal design would permit the percent- age variations in Eg* to exactly equal in magnitude the percentage variations in R. Under these con- ditions, the crystal power, as indicated in equation 1 — 401 (1) would be the same for each crystal unit. To approach such a design, g,„, as a function of Eg, would have to be known for the particular tube. Such an analysis is beyond the present dis- cussion, but the method to be used would be quite similar to that employed in the analysis of the effects of different values of crystal resistance for the Pierce circuit. As a rule of thumb, the average crystal R in the v-h-f range can be assumed to equal If the transformer-coupled oscillator is designed to drive the crystal unit at 50 per cent of its maximum rated power for the average R, there is little danger that crystal units having other values of resistance will be overdriven. For broad- band operation the crystal power will be approxi- mately directly porportional to R. When D is small the crystal power increases as R decreases as long as the percentage increase in E„* is greater than R the percentage decrease in R. With R = and p Pc = — equation 1 — 401 (1) gives an assumed value for E,’, thus (ave) Eg* = Ng* Rm or (ave) Eg = Ng Ri V Pc,„/R,., 1—402 (1) WADC TR 56-156 262 Note that equation (1) also gives the value of E» which would exist if a crystal of maximum resist- ance were driven at its rated level. However, equation (1) is less important for determining than it is for determining N^. It is assumed that a class-C value of g„, has been agreed upon. An approximate value of E^ corresponding to the chosen g,„ is thus already determined. After the value of R, is decided upon, equation (1) can serve to determine N*. 1-403. The design procedure followed so far can generally be applied to any transformer-coupled oscillator. A v-h-f pentode and a harmonic series- mode crystal unit are selected. Regulated screen and plate voltages for the tube are decided upon. An approximate class-C value of g„ and the cor- responding Eg are estimated, and average values of Pr and R are assumed. Once that O and a have been selected according to the particular require- N ments of the oscillator, the ratio can be deter- mined from equation 1 — 401 (3), and N„ from N equation 1 — 402 (1). Or, in case a particulars^ ratio is to be preferred, a and D can be determine with the aid of equation 1 — 401 (3). An alterna- tive approach, which is the one to be followed when using the table in paragraph 1-404, is to first determine optimum values for D, a, and N,,/Nj, R assume an average crystal R = and a crystal p power equal to-^, and use equation 1 — 401 (3) to determine the value of gmEg* (= I„Eg). The next problem is to determine what value of Eg will pro- duce the required value of I„Eg when the tube is operating into a plate impedance that is small rela- tive to the tube R„. The equations relating I„ to Eg in the analysis of the Pierce circuit, or rather the basic methods used to derive the equations, are applicable here if modified properly. Since some trial-and-error will be required regardless, it may well be preferable to determine the correct Eg em- pirically. The selected vacuum tube can be driven by an external generator having a variable output at a frequency near the actual frequency for which the oscillator is to be designed. The tube should operate into a smalT resistive impedance and the gridleak resistance should be the same as that to be used in the final design. Eg should be varied until the measured I,, is such that the product agrees with the value computed from equation 1 — 401 (3). With Eg approximately known, Ng can be determined by means of equation 1 — 401 (1). 1-404. The dimensionless equations listed below relate the various parameters. These equations, Section I Crystal Oscillators some of which have already been given, are also useful in determining the various circuit voltages and currents when Eg is known, or the require- ments thereof, and in comparing the characteris- tics of oscillators of different design. g... Zp = gm Zg Np“ g,.. R Ng* g... R R,/R = PL/g.n E* Pl/Pc = (D-f a) Ng (D - 1) N, (D + 1/a) N p (D - 1) N, D (a + D* Ng a (D - 1)' Ng D (a -H 1)=* N,. a (D - D* D ■ - 1 a 1 2 _ N., (D -1- Ng D (D - 1) (D -H a)* 1-M04 (1) 1—404 (2) 1-404 (3) 1—404 (4) 1-404 (5) a (a -h 1) (D - 1) 1—404 (6) 1-404 (7) The following table, prepared by the v-h-f oscilla- tor research team at the Georgia Institute of Tech- nology, lists the quantitative relations that hold for five typical designs of the transformer-coupled oscillator. D, a, and the N,,/Ng ratio are predeter- mined to provide optimum or practical operating characteristics according to five different objec- tives. A. Symmetrical circuit design that yields mini- mum values of ZpD and ZgD. By equations 1 — 399 (1) and (2), this design permits maximum fre- quency stability if the average variations in C„ and Cg are approximately equal. Low power out- put. Narrow bandwidth. B. Nonsymmetrical grid and plate impedances. Designed for optimum frequency stability when (ave) ACg = 10 (ave) aC,,. Low power output. Narrow bandwidth. C. Nonsymmetrical grid and plate impedances. Maximum frequency stability when (ave) ACg = 10 (ave) aC,. and the power output is 10 times that in design B, but the stability is less than that in designs A and B. D. Symmetrical circuit. Broad-band untuned operation. Provides small values for Z„C,, and ZgCg. Tubes require large transconductance and small input and output capacities. Z„ and Zg are one- half those in design A. Average AC,, assumed equal to average aC*. Frequency stability below WAOC TR 56-156 263 S«ctton I Crystal Oscillators average. Power output low, but greater than that in designs A and B. E. Nonsymmetrical design except that Zp = Z,. MODIFICATIONS OF TRANSFORMER- COUPLED OSCILLATOR 1-405. Four modifications of the basic trans- former-coupled oscillator are shown in figure 1-177. Circuits (A), (B), and (C) are experi- mental models that were designed at the Georgia Institute of Technology, but not in accordance with the designs given in paragraph 1-404. Cir- cuit (D) is an oscillator in actual use that has been designed specifically to operate with Crystal Unit CR-24/U without driving the crystal beyond its Military-Standard level. Figure 1-177(A) is a Af is the difference between the operating fre- quency and the series-resonance frequency of the crystal unit as measured with Crystal Impedance Meter TS-683/TSM. Figure 1-177 (B) is a circuit intended for broad-band untuned operation with a center frequency at 63 me. Satisfactory operation was obtained on a crystal plug-in basis from 53 to 73 me. The designers recommend that the phase- High power output — same as that in design C. Broad-band untuned operation. Frequency stabil- ity below average. low-power circuit intended to be operated within a band of ±2 per cent of 55 me. When tested with a number of crystal units having overtone frequencies between 50 and 60 me, the circuit, with Lp and L, adjusted to be antiresonant with C„ and Cp, respectively, at 55 me, showed the fol- lowing operating characteristics. Figure 1-177 (D) shows a slug-tuned transformer-coupled oscillator that employs a battery-operated subminiature tube. This oscillator is used in Radio Receiver- Transmitter RT-159A/URC-4. compensating network, which was selected with the help of chart I, page 446, "Network Analysis and Fe^back Amplifier Design,” Bode, be shifted from the grid to the plate circuit. Figure 1-177 (C) is a circuit designed to provide an output ap- proaching 1 watt at 50 me. Due largely to difii- culties in predicting the input resistance of the vacuum tube, the differences between the initial TRANSFORMER-COUPLED OSCILLATOR DESIGNS Parameter Value of Parameter Design A Design B Design C Design D Design E Zp/Zp 1 10 10 1 1 a 1 1 0.0985 1 0.24 D 2.414 2.414 2.414 10.65 10.65 Np/Np 1 vio 7.07 1 1.166 2.414 7.63 12.65 1.207 1.315 gn.Zp 2.414 0.763 1.255 1.207 1.315 gmZpD 5.83 18.4 30.3 12.85 14.10 gmZpD 5.83 18.4 30.3 12.85 14.0 Np^gmR 4.828 15.25 104.0 0.457 0.854 Np‘g,pR 4.828 1.525 2.10 0.457 0.63 Pl/Po 4.12 4.12 41.2 7.05 41.2 R./R 0.707 0.707 1.286 4.825 7.78 Pl 8mEp» 1.706 5.4 11.9 0.662 1.08 f (me) Harmonic Cp (M^f) R (ohms) Stability (ppm/volt) Af (kc) Pl (mw) 50 6 5 48 0.6 1.6 54 9 6 60 0.18 0.6 80 54.8 7 6 80 0.11 0.0 , 60 68.3 7 8 82 0.626 40 60 3 18 25 0.66 40 WADC TR 56-156 264 S«ction I Crystal Oscillators Figure 1-177. ModMcationt of traitsformor-touplod o$eillator for; (A) High stability, (B) UntuHod broad band. (C) large output. (D) Slug-tuned, narrow band design and the final adjustments (the latter shown in figure 1-177 (C)) were quite large. The operat- ing characteristics of the circuit are given below : Frequency stability = 0.3 ppm/volt D-C power input = 1.8 watts R-F power output = 0.6 watt Crystal power = 0.02 watt Efficiency = 33 per cent Pl/Pc = 30 Grid bias = —19 volts WADC TR 56-156 265 Saetion i Crystal Osciliotere Greanded'Grld Oscillator 1-406. The grounded-grid oscillator, or single-tube Butler oscillator (see figure 1-178) is very similar in principle to the transformer-coupled oscillator. In fact, it can be described as a transformeiv coupled oscillator in which the feedback is cathode- coupled to the input. The cathode coupling elimi- nates the need for phase reversal in either the plate or input transformer. As indicated in figure 1-178, the input transformer can be reduced to an r-f choke. For the plate circuit, an autotrans- former permits a maximum coefficient of coupling. Either a triode or pentode can be used. Although operable at lower frequencies, the oscillator is usually employed in the v-h-f range. The low input impedance reduces the effect of variations in the cathode capacitances. Unless the tube is operated class C, the crystal operates into a relatively low- resistance circuit. The power output is lower than that obtainable with a transformer-coupled oscil- lator, but for small outputs the frequency stability is quite high and the load is more readily shielded from the input. This latter feature reduces the possibility that the oscillator will operate at fre- quencies other than that of the desired mode of the crystal. The shunt capacitance of the crystal unit should be compensated by a broad-band anti- resonant inductor, or by other means. The most dependable method for use over a wide range of frequencies is to introduce mutual inductance be- tween the input transformer or choke and the plate transformer. This can be attained by wind- ing the cathode inductor on the same form as the plate transfbrmer is wound. The correct coefficient of coupling between the input and plate inductors is the one that permits the crystal to vibrate at its true motional-arm resonance. For a theoretical discussion of this mode of capacitance compensa- tion, see Eldson et al. The grounded-grid oscillator is most advantageous to use when maximum com- pactness and simplicity are desired in a low -power, broad-band, untuned v-h-f oscillator. There is the very important additional advantage that the oscil- Figure 1-178. I (C) (D) (E) Basic grcuitd»d-gr1d otcilittfcr and equivalent circuit pngreteively simplified WADC TR 56-156 266 lator can be readily designed so that both the out- put amplitude and the frequency stability are vir- tually independent of the crystal resistance. ANALYSIS OF GROUNDED-GRID OSCILLATOR 1-407. Referring to figure 1-178, it will be assumed that the plate circuit is tuned to the desired har- monic, series-resonance frequency of the crystal, that the resistance in the plate tank is negligible, that the grid current is negligible, that the auto- transformer coefficient of coupling is unity, that the cathode-to-ground capacitance is antiresonant with the cathode choke, and that the r-f current through the choke is negligible. Under these con- ditions the equivalent r-f circuit of the oscillator is that shown in figure 1-178 (B). All voltage sym- bols are treated as unsigned. The polarities shown correspond to the instantaneous polarities that hold during the positive alternations of the r-f grid voltage. Eg. The current arrows point in the instantaneous direction of the in-phase electron flow. The reactive component of the plate tank current is not represented, although in reality it is primarily the “flywheel” current that produces the voltages E, and Ej. This reactive current in the tank is that which flows through Cp and is equal to mCpEl. If Rl were reduced to zero, there would be no reactive current and the transformer would effectively short-circuit the crystal to the plate. Of course, oscillations could not exist under these conditions, if for no other reason than the fact that E, would be reduced to zero and E,, being simply the voltage across the crystal unit, would be displaced 180 degrees from the phase required for oscillations to be maintained. E, must be greater than E.. The important feature to remem- ber is that insofar as the resistive component of the current is concerned the action of the auto- transformer is the same as that of a conventional transformer when the primary and secondary cir- cuits are connected in parallel as shown in figure 1-178 (B). Note, however, that the turns ratio, N, is defined as the ratio of the total turns to the turns comprising L,. 1-408. The power fed to the transformer is simply Pl == II El = Ip Ej 1 — 408 (1) Now, El = E. -I- E, = E, -f N or El = 1-408 (2) N — 1 So Section 1 Crystal Oscillators 1 II = Ip E. (N - 1) Ip El ■ N 1—408 (3) And since Ip = Il + Ii 1—408 (4) We have Ip = (N - 1) Ip , p N 1-^08 (5) or Ip = NI. 1^08 (6) and Ir. = I. (N - 1) 1—408 (7) The ratio of the output to the crystal power is Pl/P, = II* Rl/I,," R = = Zl/R 1—408 (8) Note that for a given turns ratio, the power ratio is directly proportional to the resistance ratio. /N — IV The term Rl f — — 1 is simply the equivalent load resistance, Z,,, that the transformer presents to Ip. The total impedance across the vacuum tube is thus Zp = Rl y + R = Zl -f R 1—408 (9) 1-409. It is convenient to imagine that the ground connection is at the point G' in figure 1-178. That is, let G' be our point of reference. Insofar as the r-f circuit is concerned such a supposition requires no alteration in the currents and voltages involved, but it does simplify the visualization of the circuit charactistics. The supposition does not mean that there is no difference in the r-f potential between G' and the actual ground. With G' in figure 1-178 (B) assumed to be the ground connection, it can be seen that the tube is effectively connected as a cathode follower except that no load is taken from the cathode circuit. The crystal R is the cathode resistance, and E, is the voltage input to the grid circuit. As discussed in the analysis of the two- tube Butler circuit and as illustrated in figure 1-173, the cathode-follower type of circuit can be represented by an equivalent circuit in which the plate-circuit resistance, exclusive of the cathode resistance, is equal to — times the actual re- A* + 1 WADC TR 56-156 267 Saction I Crystal Oscillators sistance. Also, the equivalent generator voltage is — times the conventional value that would be M 1 assumed if the cathode resistance were not pres- ent. As applied to the grounded-grid oscillator, the equivalent circuit is that shown in figure 1-178 (C) . Care must be taken not to interpret the power supplied to the plate-circuit resistances as being reduced by a factor of — With u and Rp /* + 1 assumed large compared with unity and Z,„ respec- tively, circuit (C) reduces to circuit (D). LOOP GAIN IN GROUNDED-GRID OSCILLATOR 1-410. At equilibrium Gi G2 G3 G 4 where ^2. . .B. . ..^ . ^ = 1 R 2 1^0 1-410 (1) G = = R ' E 2 ■ Rp -h Zl + R (m + 1) J_ R Sm 1—410 (2) G2 = E,/E„ = Z,yR = 1^10 (3) G3 = El/Ei = N/(N - 1) 1—410 (4) G4 = Ej/El = 1/N 1—410 (5) Equation (2) simply represents the gain of a cathode follower. It can be derived from the equiv- alent circuits in figure 1-178 in a manner similar to the cathode follower gain derivation in the analysis of the two-tube Butler circuit. In com- bining equations (2), (3), (4), and (5), we have Gi G2 G3 G4 (Rp -h R -f mR) ^ 1 Ri. (N - 1) (M + 1 - N) 1—410 (6) If the approximate value for G, can be assumed, the gain equation can be expressed as 1 Rl (N - 1) x> _ Zc x> gp, N - 1 1—410 (7) ACTIVITY CONSIDERATIONS IN GROUNDED-GRID OSCILLATOR 1-411. From equation 1 — 410(7) we can predict WAOC TR 56-156 268 the relative activity to be expected with different load and crystal resistances. When oscillations first start, the term l/g,„ is a minimum, 'fhe amplitude increases until g„, is reduced by the increase in grid bias to the value that makes equation 1 — 410 (7) hold. The greater the difference between the initial, zero-bias value of gm and the equilibrium value, the greater will be the final amplitude. The initial g„ of the tube should therefore be as large as possible if a maximum output is desired. For a given crystal R, the output is increased by increas- N — 1 ing Ri, and making the ratio — a maximum, which occurs when N = 2. Assuming that N = 2, i.e., that the transformer coil is center-tapped at the connection to the crystal, equation 1 — 410(7) becomes 1/gm - R1./4 - R = Zl - R 1—411 (1) In equation 1 — 408 (8), it was found that the power ratio, to which we shall assign the symbol r^ = ^Y is equal to Zi,/R. On substituting the value rR for Zl in equation (1) we find that r = 1 -I 1-411 (2) gm R By equations (1) and (2), which hold only when N = 2, we see that increasing the amplitude by increasing Ri. results in simultaneously increasing the power ratio. Also, note that the minimum power ratio can be predetermined as a function of the class-A value of g„ and the maximum per- missible R of the crystal unit. If variations in R from crystal unit to crystal unit are not to have a large effect upon the output amplitude, ZL^=-^^must be large compared with the maxi- mum crystal R. In other words, the minimum power ratio should be as large as possible consist- ent with the requirements of frequency stability. CRYSTAL DRIVE-LEVEL CONSIDERATIONS IN GROUNDED-GRID OSCILLATOR 1-412. If P,„, is the rated maximum power of the crystal unit, the maximum permissible I„ for a given R is (max) Ip = (max) E, g„ = V Pom/R 1—412 (1) Consequently, (max) El = (max) Ip Z,. = Zl V Pcm/R 1—412 (2) By equations 1 — 410 (4) and (5) , ^ t:, (max) E, N Zl N V P.m/R (max) E, = = — N_i and (max) E2 1-412 (3) (max) El/N = Assuming that N = 2, we have 1—412 (4) (max) E|. = (max) El = (max) E2 = 4 1—412 (6) The maximum permissible excitation voltage of the tube with a center-tapped transformer is (max) E, = (max) I, £ — ga (max) E2 — (max) 1-412 (6) IV Equation (6) is to be interpreted as giving the maximum permissible E* for a given value of R. The smaller the value of R, the larger will be the permissible value of E,. For v-h-f crystal units in which the shunt capacitance of the crystal is not compensated the minimum R encountered may be on the order of Rm/5, where R^ is the rated maxi- mum. However, with capacitance compensation all values of R will be less than R„, and the minimum may well be on the order of Rm/9 or less. The plate characteristics of the vacuum tube and the electrode voltages must be such that an excitation voltage equal to the maximum permissible Ej, as defined by equation (6) , does not cause an effective Ip greater than \/I’cm/R- With a sharp-cutoff, grid- leak-biased tube operating into a plate impedance that is small compared with Rp, the effective Ip remains essentially constant as the peak-to-peak amplitude of E, is increased from a value equal to |E„| to a value equal to 2 |Em|, where E„ is the cutoff bias. As discussed in paragraph 1-312 in connection with the Pierce circuit, the crest ampli- tude of Ip between the crest values of E, equal to and |Eco| remains approximately equal to where Ibm is the zero-bias plate current. There is a small maximum approximately equal to 0.54 Im (see equation 1 — 812 (21)), but for all practical Section I Crystal Oscillators purposes it can be assumed that Ip is constant for all values of gm between class-A and class-B opera- tion. Since the effective gm is equal to Ip/E,, the doubling of Ep without changing Ip is equivalent to halving g,,. If oscillations can be maintained at all, the slightest tolerance allowed in gp, ensures that the amplitude will build up until the excita- tion voltage overlaps the lower bend in the EJp curve. Thus, for a sharp-cutoff tube the niinimum equilibrium Egp, ( = \/2Eg) will very nearly equal Since l/g^ = Zl — R when N = 2, any value of Zl greater than 2R„, can ensure that the r-f plate current, and hence the output, will be the same for all values of R falling within the crystal specifica- tions. All that need be done is to design the circuit for class-A operation on the assumption that R = R,n. This requires the use of a sharp-cutoff vacuum tube with tube voltages such that Il„ < 2^1^^ 1-412 (7) and the design of the load and transformer net- work such that ^ - R„ 1—412 (8) Ibm N - 1 Under the conditions defined by equations (7) and (8), a crystal having a maximum R will be driven at or under the rated maximum drive, depending upon whether Ib„, is equal to, or less than, the value specified in equation (7) . If the crystal unit is re- placed by another of lower resistance, the crystal current and the output will remain essentially the same. Since E^ will also be unchanged, the increase in the excitation voltage will be entirely that due to the decrease in the voltage across the crystal. The driving power of the crystal will be directly proportional to the crystal R. FREQUENCY STABILITY OF GROUNDED-GRID OSCILLATOR 1-413. In the equivalent circuit shown in figure 1-178 (D), it can be seen that the effective resist- ance, Rc, of the crystal circuit is ( 1- R ). But, \«m I — R* thus If N = 2, Rc = Zl, = Rl/4 1—413 (2) WADC TR 56-156 269 Section I Crystal Oscillators Equations (1) and (2) assume that the mu of the tube is large compared with unity, and that R|. >> Zi,. From equation 1 — ^241 (2), the frac- tional change in frequency required to compensate a small change, dtf, in the feed-back phase is dw Rc d0 Zi, Ri, do T “ 2 VTVC " 2(N - DVXTC “ 8VTV(? 1-413 (3) where L and C are series-arm parameters of the crystal, and equation (2) is assumed to hold. Of significance is the fact that for a given Zl, the frequency stability is independent of the resistance of the crystal. However, must always be greater than R, else the conditions for oscillation as defined by equation 1 — 410 (7) cannot hold. Thus, although the frequency stability can be con- sidered independent of R for a given Zi,, the effec- tive Q of the crystal circuit must always be less = ^— jof the crystal. The larger the g,„ of the tube, the more nearly can this limiting value for the effective Q be reached. since the more nearly can be made to approach R,„ in magnitude. But a large g„ must be accompanied by a low cutoff voltage for the tube, else equations 1 — 417 (7) and (8) cannot be made to hold and the crystal will be overdriven. Unfortunately — yet not unexpectedly — the re- quirements for maximum output are the reverse of those for maximum frequency stability. If the only frequency-stability problem were to maintain the circuit Q as high as possible, the output could be increased without decreasing the stability, by making both Rl and N large. This could permit an increase in Z|. without affecting the value of On the other hand, if dm/u in equation (3) is to be kept small, not only ^ j^^ -y^but also Ae must be kept to a minimum. Because the input impedance of the tube is very low, changes in the cathode capacitance have a negligible effect on the feed-back phase. The principal variations in the phase are due to changes in the plate and load capacitance. To reduce these effects to a minimum, Zi, must be as small as possible. Its smallest per- missible value will occur when /N-IV ( N« )* IS a maxi- mum; that is, when N = 2. Letting N = 2 and R,, = 8R„„ Zi, will equal Ri,/4 = 2R„.. The crystal- circuit Q for all values of R will then be one-half the minimum crystal Q to be expected for the par- ticular type of crystal unit. The output will ap- proximately equal 2Pp„, for all values of crystal R. Much larger outputs can be obtained without greatly reducing the frequency stability, by the use of remote-cutoff tubes. With these tubes the r-f plate current can be made to vary inversely with the square root of the crystal resistance. Under these conditions, it would be the crystal power that remains constant and the output power that varies with R. If Pl = 2P<.m when R = R„, Pi, = 18P,.„, when R = Rm/9. DESIGN PROCEDURE FOR GROUNDED-GRID OSCILLATOR 1-414. The design procedure depends considerably upon the special requirements to be met by the circuit. As a concrete example assume that a low- power, 50-mc oscillator requiring a minimum of circuit components and an output amplitude that will not be greatly affected by a replacement of the crystal unit with another of the same frequency is desired. The grounded-grid oscillator is prob- ably the best suited for such a purpose. Assume further that a frequency tolerance of itO.Ol per cent is required without temperature control for all temperatures between —40 and -)-90 degrees centigrade. Crystal Unit CR-24/U with a fre- quency tolerance of ±0.005 per cent between —56 and -f-90 degrees centigrade should be able to pro- vide the required stability. So also will Crystal Unit CR-23/U, but the former unit is mounted in the coaxial holder, the HC-IO/U, which is gen- erally to be preferred because its lower inherent shunt capacitance should permit a higher average Q. There is no guarantee of this, since the maxi- mum Co is 7 ^f in each case; however, the CR- 24/U employs the 5th harmonic and the CR-23/U the 3rd harmonic (thinner crystal) for the 50-mc frequency. The greater CR-24/U L/C ratio should more than offset its slightly higher R^. Neverthe- less, a check should be made to see if crystal units of either type having the desired frequency are currently being manufactured or have been manu- factured in the past. If not, serious consideration should be given to the possibility of employing a different frequency. The cost of the crystal unit will be less if it is already in production, and the risk that an undue amount of experimentation will be required to produce a crystal unit that meets the military standards at an unexplored frequency can be avoided. If the crystal unit is expected to withstand considerable mechanical shock, the CR- 24/U must be used, regardless. 1-416. Assume that a 50-mc CR-24/U crystal unit WADC TR 56-156 270 has been selected. According to Military Standard MS91380, Rm = 75 ohms and P.„, = 2 mw. By equation 1 — 412(1) (max) Ip = 10®^ ^^^ = 6.2 ma According to equation 1 — 412(7) Ibn. < 2VX X 5.2 = 14.7 ma Assume that the most available tube is the 6AU6, sharp-cutoff, miniature pentode. Operated at 250 plate volts, a Screen voltage of 140 volts provides a zero-bias plate current of approximately 15 ma. The cutoff bias will be approximately —5 volts. Assuming a value of N = 2, by equation 1 — 412(8) Zl = 75 -f = 410 ohms and Rl = 4 Zl = 1640 ohms The power ratio when = R^ will be The power output for all values of R will be Pl = (min) r Pom = 11 mw The crystal unit will operate into an effective re- sistance equal to 410 ohms. The effective gm of the tube will vary from approximately 3000 ^imhos, when R is a maximum, to approximately 2500 fonhos, when R is a minimum. If greater frequency stability is required, Ri, can be decreased by ap- Sectien I Crystal OMillatort proximately three-fourths, so that Zl = 300 ohms. With this value of Z,„ when R is maximum gm will be 4450 /imhos. By increasing the screen volt- age to 150 volts or slightly greater, an r-f plate current very nearly equal to the maximum permis- sible for Rm can be attained. As R is decresised Ip and Pl increase somewhat, but the crystal unit will not be overdriven. MODIFICATIONS OF THE GROUNDED-GRID OSCILLATOR 1-416. Figure 1-179 shows four different designs of the grounded-grid oscillator which were built and successfully tested at the Georgia Institute of Technology. Because of the high initial transcon- ductances, 0.011 fimho for the 6J4 and 0.009 for the 6AH6, the oscillation amplitude of these cir- cuits would drive the average Military Standard crystal unit beyond the recommended maximum level. This does not mean that a standard crystal unit will necessarily be in danger of being shat- tered by the circuits shown, but that the frequency, resistance, and freedom from spurious modes could not be guaranteed by the test standards. To employ the circuits illustrated in figure 1-179, dif- ferent tubes, or plate-supply voltages may need to be used. 1-417. Figure 1-179 (A) is a narrow-band oscilla- tor with the load connected across the secondary of the plate transformer. Except for the fact that the input “transformer” (Lk, having a 1 : 1 voltage ratio) provides no phase reversal, the circuit is very similar to that of the basic transformer- coupled oscillator. Rl, connected as shown, is equivalent to a load resistance of N'^Rl connected across Lp. The variable capacitance is for tuning out the transformer leakage inductance. Lk is anti- resonant with the cathode capacitance, and the Figure I- 1 79. Modtflcafions of grounded-grid oscillator. (A) Narrow-band circuit 271 WAOC TR 56-156 Section I Cryctol Oecillators Figun 1-179. ModMieationa ttf grot$ndmd-gHd escll/otor. (B) Brooft-band eircuH. (C) Circuit for compontatlng crystal capacitanco by mutual Inductanco. (D) HIgh-offIcloney class-C circuit WADC TR 56-156 272 cathode resistance broadens the tuning. The cir- cuit was tuned for operation at a center frequency of 55 me. The plate transformer consists of 12 turns of AWG No. 26 PE wire wound on a Miller Section I Crystal Oscillators type 69048 slug-tuned coil form and tapped at 4 turns. The circuit operates class B. The operating data for several different crystal units is given below. f (me) Harmonic R (ohms) Stability (ppm/ volt) n^m Pi. (mw) 48 3 BH 45 0.42 3 65 50 3 35 0.20 1.5 103 54 3 45 0.11 9 78 58 . 7 32 0.14 0.6 28 60 3 13 25 0.24 1.6 40 The frequency stability is measured in average parts per million per volt when the voltage is changed by 50 volts. Af gives the deviation ob- served between the series-resonance frequency, when measured with Cl Meter TS-683/TSM, and the actual oscillator frequency. It would seem that the 54-mc crystal, which should show the smallest value of Af, was influenced by a spurious mode. 1-418. The circuit in figure 1-179 (B) is designed for broad-band untuned operation. Lp consists of 10 turns of No. 30 PE wire, tapped at 3.3 turns and wound on a 0.4-inch-diameter form. The ter- tiary winding, Lr, consists of 17 turns on a 0.24- inch form that can be slipped inside the L„ core by a screw adjustment. The circuit is first tuned with a 1000-ohm load connected directly across Lp with Lt open. Next, with the circuit connected as shown, the coupling between Lt and Lp is adjusted until the same grid current as before is obtained. All tuning adjustments were made at 57.5 me. The performance data of this circuit for several differ- ent crystals is given below. f (me) Harmonic R (ohms) Grid Ic (/lamp) Stability (ppm/volt) Pl (mw) 48 3 14.5 25 58 6 41 50 3 11 28 85 -0.6 45 58.31 7 8 80 60 0 52 65.31 7 10 80 65 0.15 0.2 46 66.65 5 4 65 90 0.21 mSm 50 67.2 7 14 80 62 0.07 ■1 47 1-419. The circuit in figure 1-179 (C) is designed to compensate the capacitance of the crystal unit by mutual inductance between the plate and cath- ode inductors instead of by a shunt inductor as in circuit (B). L, consists of 9 close-wound turns of AWG No. 30 PE wire, tapped at 2.5 turns; Li, consists of 4.5 turns of AWG No. 30 PE wire on a thin spacer. The proper coupling adjustment is obtained by substituting a capacitance equal to C, in place of the crystal and adjusting the circuit to oscillate at the true series-resonance frequency of the motional arm, but only after L. and L^ have separately been adjusted to resonate with Cp and Ck, respectively, at their computed resonant fre- quencies («>,’ = wb* = 7 ^) • It can be shown L.Cp J-<b^k WADC TR 56-156 that Lp N,^ Lp (1 - M^) (Lp -H N,* L„) Lk Na^ Lp (1 - M*) (Lk -h N/ Lp) 1—419 (1) 1—419 (2) and Lp U/N.^ (Lp -h N,^ Lp) (Lp -H Lk/N**) 1—419 (3) where Lp and Lk are the values of the plate-to- ground and cathode-to-ground inductances, respec- 273 Section I Cryttal Otrillaton tively, that would occur if there were no coupling between them, L„ is the imaginary shunt induct- ance that would be required to antiresonate C„, M is the coefficient of coupling between the plate and cathode inductors, and N, and Nj are the plate and cathode turns ratios, respectively. Nj is simply equal to unity in the circuit shown. The perform- ance data for the circuit is as follows: f = 58.31 me stability = 0.28 ppm/volt Pl = 90 mw Af = operating freq minus tested series- resonance freq = —100 cycles Af when Ck increased from 10 to 13 /i^f = —60 cycles Af when C„ increased from 8 to 11 /»/tf = —45 cycles Af when Cp increased from 6 to 6.5 nid = —90 cycles 1-420. The circuit shown in figure 1-179 (D) is designed for high-efficiency operation as a small class-C power oscillator. L,, consists of 20 turns of AWG No. 28 PE wire wound on a 0.25-inch coil form and tapped at 1 turn. Lj, is a 10-turn, 0.25- inch-diameter coil of AWG No. 28 PE wire, tapped at 5 turns. The observed performance data for this circuit is as follows; f = 50 me load voltage = 9 volts grid bias = — 10 volts load power =1.9 watts crystal power = 0.08 watts frequency stability = 0.6 ppm/volt plate dissipation = 3 watts efficiency = 63 per cent concerned. The grounded-plate oscillator can be designed for larger outputs by providing a step-up transformer in the cathode circuit and removing the r-f voltage from the gridleak resistor, as is shown in figure 1-180 (B). This permits the cath- ode-follower to operate into the same output im- pedance but with a greatly reduced load resistance across the crystal circuit. The output per milliwatt of crystal power is thereby increased. Increasing the power output in this manner makes the oscil- lator more critical to design and adjust so as to prevent free-running oscillations, particularly if the tube is to be operated class C, where the effec- tive input impedance becomes more or less un- predictable at very high frequencies. 1-422. The over-all gain equation of the oscillator in figure 1-180 (A) is MNRaZk (R Rj) (Rp -)- Zk -f- pZk) 1-422 (1) where Zk is the total effective resistance between the cathode and ground. Assuming that the resist- ance presented by the transformer is equal to Rg/N* and is much greater than R„ we have R] (R -i- R 2 ) Rj + R + R2 1—422 (2) The effective resistance into which the crystal operates is Rp = Zk' 4- R -h Rj 1—422 (3) where Zk' is the output impedance of the cathode follower as faced by the crystal. If is very large compared with unity. The Grounded-Plote Oscillator 1-421. The vacuum-tube circuit of the grounded- plate oscillator shown in figure 1-180 is essentially the same as the two-tube Butler oscillator except that a step-up transformer replaces the grounded- grid amplifier of the Butler circuit. The gain of the Butler grounded-grid tube is thus replaced by the gain, N, of the transformer in figure 1-180 (A) . The grounded-plate oscillator is most advanta- geous when used in the electron-coupled form, as shown in figure 1-180 (C), where the plate circuit can be tuned to provide frequency multiplication. Otherwise, the larger output of the basic trans- former-coupled circuit or the greater simplicity of the grounded-grid circuit make these oscillators preferable to the grounded-plate design insofar as obtaining the same order of frequency stability is 2'‘' = 1 , p 1-422 (4) 1 + gn. Rl For crystal resistances on the order of 75 ohms or smaller, R, and R, can also be approximately 75 ohms each. Values of R, = 68 ohms, R, = 100 ohms, Rg = 200K, and N = 9 have been recom- mended for use with a 6J4 triode. The shunt ca- pacitance of the crystal unit, as well as that of the grid and cathode, can be compensated if need be by conventional antiresonant inductors. To be pre- ferred is the method described in the discussion of the two-tube Butler circuit — designing the cir- cuit so that gm Zk = ^ 1--422 (5) Uk WADC TR 56-156 274 Section ! Crystal Oscillators 8 + figun 1-181. Traitthron crystal oscillator Transitron Crystal Oscillator 1-423. The transitron oscillator (see figure 1-181) operates by virtue of the negative transconduc- tance between the suppressor and screen grids of a pentode. The total cathode current of the pentode is little affected by variations in the suppressor voltage, being primarily a function of the poten- tial between the screen and cathode. However, as the suppressor voltage is made more negative, the fraction of the total space current diverted to the screen circuit is increased. The screen voltage therefore tends to follow the suppressor voltage. By connecting a resonant feed-back network be- tween the screen and suppressor, oscillations can be maintained and no phase reversal is necessary. The principal advantage of this circuit is its sim- plicity and its ability to oscillate with series-mode crystals having comparatively high series resist- ances. It can be employed in the v-h-f range, but unless the crystal resistances are expected to be abnormally high, the relatively large electrode ca- pacitances and the small transconductance make the performance inferior to that of the trans- former-coupled oscillator. 1-424. When the circuit is used with high-resist- ance crystals it is very important that the crystal shunt capacitance be properly compensated, in order to eliminate the possibility of free-running oscillations. As has been demonstrated by W. A. Edson with the aid of Nyquist diagrams (graph- ical representations of the over-all loop gain and phase rotation as the frequency is varied from 0 to 00 ) , the circuit can be designed to permit only one mode of oscillation if, treating g„ as unsigned. gL^/ Ci-hC, Cl C,\ R. \ R„ ^ R, R, / C„ 1—424 (2) The condition implied by equation (1) when gm is its maximum possible value means that the loop gain is insufficient to start or maintain oscillations at any frequency unless R, is effectively decreased (such as being bypassed by the series-resonance R of the crystal) so that the left side of the equation is greater than or equal to unity. Equation (2), when satisfied, means that a zero phase shift in the feedback can occur at only one frequency. Thus, if the circuit is tuned for operation at the desired series-mode frequency of the crystal and equations (1) and (2) are satisfied, spurious oscil- lations will not be possible. 1-425. Note that Co is effectively increased by the suppressor-to-screen capacitance, so that Lo must be smaller than would otherwise be the case. C, and Cg are simply distributed capacitances to ground. Each of the three parallel combinations are antiresonant at the crystal frequency, so L,C, = L,Co = LgCg = Assuming that the antiresonant circuits have impedances Rl, R and Rg, respectively, then if Ig, is the r-f screen cur- rent, the voltage across the load is _ Ig2 Rl (R + Rg) Ri, -H R + Rg The r-f suppressor voltage is 1—426 (1) Km Hl Rg ^ , Ro -f Rl -t- Rg 1—424 (1) _ Eg2 Rg _ 1,2 Rl Rg ** R -h Rg Rl + R + Rg 1—426 (2) If we assume that E,, is small compared with the and WADC TR 56-156 276 Section I Crystal Otciliators d-c screen voltage, and define the suppressor-to- screen transconductance as the change in screen current per change in suppressor-to-cathode volt- age — not per change in the suppressor-to-screen voltage — the gain conditions for equilibrium are, by equation (2), Eg3 1 _ Rl 1*2 Sm Rl + R + Rg 1-425 (3) Of the vacuum tubes available, the 6AS6, which has a suppressor-to-screen transconductance of 1600 fimhos, is probably to be preferred. With this tube, oscillations can be maintained with crystal units having series resistances of well over 1000 ohms. Although oscillations can also be main- tained with large values of Rl and Rg, these re- sistances should be kept as small as practicable so as not to unnecessarily degrade the crystal Q and reduce the frequency stability. The transitron oscillator is also quite useful at low frequencies, particularly with high-resistance crystal units. When a fundamental-mode crystal element is em- ployed, the tuned circuits may not be necessary ; but to avoid the possibility of free-running oscil- lations or unwanted crystal modes, at least the screen circuit should be broadly tuned. (See para- graph 1-590 for discussion of negative-resistance limiting of transitron circuit.) Impedance-lflveriinq Crystal Oscillators 1-426. Impedance-inverting oscillators employ a network similar to that shown in figure 1-182(A) , to permit conventional lower-frequency oscillators to be operated with crystal control in the v-h-f range. A number of these oscillators were de- signed and tested at the Georgia Institute of Tech- nology under the direction of Mr. W. A, Edson. The discussion to follow is based on the final re- port of this research. The impedance-inverting network is designed to behave as a quarter-wave line having a characteristic impedance, Z« = uLi = . With this design, the network ZJ always appears as an inverted Z, equal to Z. = where Z. is the series-arm impedance of the crys- tal. If Z. = 0, Co is shorted out and Zg is infinite. (L, is assumed to have a zero loss.) If Z, is in- finite, Lj is series-resonant with Co, and Zo = 0. If Z. = Zo, the network appears as an infinite line with Zo = Zo. When Z, is a small inductive react- ance, Zn is a large capacitive reactance, and vice versa. With Zo > > R of the crystal, the network Figur* 1-182. Impedance-inverting osciflator circuits. (A) Baak impedatKe-inverting network serves to invert the crystal resistance to a high impedance equal to Zo*/R. For a given C„, maxi- mum frequency stability is to be had under the quarter- wave line conditions (C„ = C,), but, if desired, higher impedances can be had by making Co less than C*, or by reducing the effective values of Co and C„ with the use of shunt inductors. The shunt inductances, however, should be consider- ably larger than the values required for antireso- nance at the crystal frequency. With «> equal to the series-arm resonance frequency, and L, z= Z„ appears as an antiresonant re- <!> l-'o W sistance when the series arm of the crystal is reso- nant and Z, = R. At frequencies well removed from crystal resonance, the crystal behaves simply as a capacitance, C„, .so that the network has a sec- ond antiresonant frequency, the square of which C 4- C is wj’ = -. To ensure that this second fre- quency is damped out, a resistance equal to Z, can be connected across the crystal unit. 1-427. Even though the equivalent impedance-in- verting network is design^ to be antiresonant at approximately the crystal frequency, the operat- ing frequency may well require that the crystal network facing the actual terminal connections be reactive if the necessary phase reversal is to be accomplished. For example, it is necessary that the actual plate-to-grid network appear inductive when used in the Pierce circuit. In the Pierce cir- cuit the fundamental modification introduced by the impedance-inverting circuit is simply the ad- dition of an inductor having a reactance uL, = -7:^ in series with the crystal. It can be imagined that the reactance of the inductor replaces the X, of a WADC TR 56-156 277 Section I Crystal Oscillators parallel-resonant crystal unit* and the low series- resonant R of the crystal approximately replaces its parallel-resonant value, R,.. C„ of the network is C|,„ of the vacuum tube, if we view the circuit literally. By this interpretation, C„ is not anti- resonant with the inductive branch, but must offer a higher impedance than does the inductor at the operating frequency. On the other hand, if the entire external circuit is viewed in toto by the negative-resistance method, which is the imped- ance-inverting interpretation, C„ appears as an equivalent capacitance equal to 0,,^ plus the addi- tional amount required to make the network anti- resonant. Since this latter interpretation can be employed to illustrate any oscillator circuit that contains a crystal connected in series with an in- ductance L, = l/<o*C„, the presence of the series inductance alone could be sufficient to define an impedance-inverting oscillator. The inverted im- pedance, Z„, is related to the impedance of the inductor and crystal branch as the PI of a crystal is related to the equivalent impedance of the crys- tal unit. Although these questions are somewhat academic, for some readers it may be more helpful to interpret the network in figure 1-182 (A) as an impedance-converting circuit rather than as an inverting circuit. In the transitron circuit, the net- work is directly used to invert the crystal R to a higher effective resistance, but in other applica- tions the designer may prefer to treat the actual network as an equivalent X, and R* of a parallel- mode crystal unit, transferring the equivalent im- pedances directly to the equations of the basic parallel-resonant oscillators. 1-428. There are two significant advantages to the impedance-inverting type of design. One is that the conventional parallel-resonant circuits can be operated with excellent frequency stability in the v-h-f range. Another is that by using series-mode crystals at the fundamental frequencies, the design restrictions regarding the parallel-resonant type of crystal unit can be avoided. No data is avail- able, but experimentation may show that even in the fundamental-frequency range larger outputs can be obtained with an inductor and a series- mode crystal without degrading the over-all fre- quency stability. The chief disadvantage of the impedance-inverting network is that it cannot be used for broad-band untuned operation. IMPEDANCE-INVERTING TRANSITRON OSCILLATOR 1-429. An experimental 50-mc impedance-invert- ing transitron oscillator is shown in figure 1-182 (B). In this circuit, the network, consisting pri- marily of L,, Cn, and the crystal, is adjusted to present a resistive impedance between the screen and ground. Since Ri, is very large compared with the crystal R, it can be assumed that Rt, is effec- tively connected in parallel with the antiresonant network. C„ includes a 3 — 12 /t/if padding capaci- tor adjusted at 5 /i/i»f, the screen-to-ground capaci- tance, and the suppressor-to-plate capacitance (the latter is added because the screen is prac- tically bypassed to the suppressor and the plate is at r-f ground). E*, in this circuit can be consid- ered equal to E,j. Thus, the condition required for oscillations to build up is simply that 1/gm be smaller than the actual plate-to-ground resistance. At the plate voltage used, the initial gm is approxi- mately 1500 /imhos, so 1/gm = 667 ohms. Ignoring the suppressor-to-ground resistance, the screen operates into an impedance of Ri.Z„/(Ri. -f- Z„) = 1350 ohms, where Z^ = XcnVR. The margin of gain is therefore on the order of two to one. The power delivered to Rl was observed to be 15 mw. The frequency deviation was measured at 0.1 ppm/ volt. When Ri was replaced by a 40,000-ohm re- sistor, the frequency deviation was found to be only 0.004 ppm/volt. Although the power output is low, the extraordinary independence of the fre- quency under variations in the supply voltage marks the impedance-inverting transitron oscil- lator as the most stable to use in the v-h-f range. One of the chief reasons for this stability is very probably the fact that the r-f screen current need contain no reactive component. The impedance- inverting network, as faced by the screen, can appear as a pure resistance. IMPEDANCE-INVERTING PIERCE OSCILLATOR 1-430. A 50-mc impedance-inverting Pierce oscil- lator is shown in figure 1-182 (C). This circuit supplied 70 mw to the 1500-ohm load, and had a frequency deviation of 0.6 ppm/volt. Note that the total Cb is equal to the total C„. The antiresonant Cn for the inductive branch of the impedance-in- verting network is thus very nearly (Cp* -j- Cp/2), which in turn is equal to Cp. This value of Cn neg- lects the equilavent negative capacitance due to the reactive component of the r-f plate current. Viewed only as an impedance-converting network connected between the plate and grid, Cn = Cpg, and the network appears as an inductive reactance numerically equal to 2/o,Cp or 2/wCg. The upper useful limit of this type of circuit is approxi- mately 100 me, 1-431. Figure 1-182(D) shows an electron-coupled modification of the impedance-inverting Pierce WADC TR 56-156 278 Section I Crystal Oscillators RFC R| (B) Figure f-182. Impedanee~inverting oeeillator eircuits. (B) Impedance-inverting transitron oscillator. (C) Impedance-inverting Pierce oscillater. (D) Impedance-inverting electron-coupled Pierce escillater oscillator. The frequency of the oscillator circuit is virtually independent of the tuning adjustments in the plate circuit. With the plate circuit tuned to the 1st, 2nd, 3rd, and 4th harmonics succes- sively, the power supplied Rl was found to be 400 mw at 54 me, 225 mw at 108 me, 50 mw at 162 me, and 10 mw at 218 me. The frequency stability is approximately the same as that of the triode circuit in figure 1-182 (C). The upper frequency limit of the grounded-screen circuit in (D) was found to be 70 me. WADC TR 56-156 279 Section I Cryctal 0*cillatera (E) Figure 1-187. Impedanee-InvertiHg oatillater cinuita. (i) Impedenn-invrting Miller otclllafer IMPEDANCE-INVERTING MILLER OSCILLATOR 1-432. Figure 1-182 (E) is an experimental design of a 50-mc, impedance-inverting, Miller oscillator. The oscillator is designed so that the C, of the equivalent negative resistance circuit is equal to Co. Assuming that the reactive component of the plate current is negligible, Cn = Co = Cg -f- C„ where C, is the equivalent capacitance of Cp, in series with the parallel combination of Cp and Lp. Cl is thus given by the equation Cpg (1 - 0.^ Lp Cp) I (Cp -h Cp,) 1-432 (1) Cp, and Cp are fixed by the tube capacitances, and C, is equal to C, — C„ so the solution of equation (1) requires a definite value of Lp, which in cir- cuit (D) was found to be 0.744 ^ih. The circuit supplies Rl with a power output of 0.5 watt for a crystal drive of 0.07 watt. The frequency devia- tion was found to be 0.6 ppm/volt. Grounded-Cathode Two-Stage Feed-Back Oscillator 1-433. The two-stage feed-back oscillator (see fig- ure 1-183) is used primarily for high-resistance, series-mode crystals operating at fundamental frequencies not higher than 500 kc and usually below 300 kc. The design is rather straightfor- ward. V, and V, are tubes of the same type and can bb contained in the same envelope. Although pentodes should permit slightly greater frequency stability, triodes are quite satisfactory for most purposes. Since Vj alone can provide the neces- sary phase reversal, both tubes can operate into resistive loads. The Vj plate circuit is thus tuned to the crystal resonance frequency. The proper adjustment of Cp is indicated by a maximum read- ing on the meter, M. R, is connected across the Lp-Cp tank, to broaden the tuning and reduce the frequency effects of variations in the Vj plate ca- pacitance. The resistance, R„ of the crystal circuit is approximately R -f- 2R,. On the assumption that R = Rp, (the maximum permissible crystal resistance), R, should be made as small as pos- sible consistent with stable oscillations. This is desirable in order for the effective Q of the crys- tal circuit, and hence the frequency stability, to be maximum. 1-434. The loop-gain requirement for equilibrium is G. G* Ga = . |si = 1 1-434 (1) JJigi JJipi JJjpa where G, = = g„. Ra 1^134 (2) L,i * Ep, R -I- 2R, and C = 5si = Epa R -I- R, 1-434 (3) 1—434 (4) Equations (2), (3), and (4) assume that V, and Vi operate into plate impedances approximately equal to R, and » respectively, and that these impedances are very small compared with the Rp of the tubes. Combining equations (2), (3), and (4), we find that at equilibrium, the tube transconductances are such that Gi Gj G* g»l 8m2 Rl* R 2 R 2Rj 1-434 (6) WADC TR 56-156 280 Section I Crystal Oscillators figun 1*183. Grounded-cathode twe-ttage feed-back etcillator Assuming that ga,, gm* = g„*, where g„ is the nom- inal class-A transconductance of the V, and V, tsrpe of tube, and that R = Rm, we can select values of R, and R^ so that equation (5) will equal 1.5. This provides a 3-to-2 margin of gain, which should be sufficient to ensure operation with all but completely defective tubes. The cathode re- sistors can be selected so that the amplitude of oscillations does not overdrive a crystal of maxi- mum R. An alternative, and possibly a simpler approach, is first to select a cathode resistor for V„ with the intention of operating that tube at a fixed class-A bias. The gain of the Vi stage can then be treated as a predetermined constant and the V, stage designed to provide the neces- sary limiting by gridleak bias. The class-A gain of the V, stage must be sufficient to permit oscilla- tions when the crystal unit has a maximum re- sistance, and the excitation current must not be sufficient to overdrive the crystal unit when the crystal resistance is a minimum. If desired, a parallel-mode crystal unit connected in series with its rated load capacitance can be substituted for a series mode crystal unit. Such operation in- creases the average effective feedback resistance, but the presence of the capacitor can reduce the tendency of the circuit to oscillate at unwanted frequencies. MODIFIED TWO-STAGE FEED-BACK OSCILLATOR 1-435. A modification of the two-stage, feed-back oscillator to reduce the higher harmonics and thereby improve the quality of the sine-wave out- put for sync control is shown in figure 1-184. It can be seen that the tuned tank, undamped, is con- nected in the plate circuit of V, instead of that of Vi as is conventionally done. The output, E„ is taken from a different part of the tank in each of the three circuits represented. The non-bypassed cathode resistors are inserted for their degenera- tive effect on the higher harmonics and parasitic frequencies. They also reduce the effective input capacitance of the tubes. It would seem that the degradation of the crystal Q is somewhat large. The Miller effect in V, is probably significant in determining the impedance that the crystal faces — certainly so in circuit (C), where R, is one megohm and C. is inserted to increase the plate- to-grid capacitance by 25 /^if. However, the chief purpose of Cg is to serve as a neutralizing ca- pacitance for all free-running oscillations where the crystal unit would behave as a capacitance, C,. Celpitts Oscillators ModHiocI for Crystal Control 1-436. Figure 1-185 illustrates a number of spe- cial-purpose circuits which are basically Colpitts oscillators modified for crystal control. Circuits (A), (B), and (C) are conventional Cl-meter oscillators (see paragraph 1-220) . The tank in- ductance, equal to 2L„ is split into two equal in- ductances, Lia and L,b. Each of the variable inductors in circuit (A) actually represent seven fixed inductors which can be connected into the circuit by a range switch. The capacitors C, (A and B) are continuously variable, and are so ganged that Cia is always equal to C,b. Circuits WADC TR 56-156 281 Saetion I Crystal Oscillators Fig. Equipment Purpose F, , CR Hi Rr Rs R« Rs (A) Range Calibrators TS-102/AP and TS-102A/AP Crystal con- trol of 500-yd marker and sync pulses 327.8 Sig C Stock No. 2X62- 327.8; WECo No. D-168342 1.8 10 18 1000 18 (B) Calibrator TS-ld/APQ-5 Crystal con- trol of 1000-ft marker and sync pulses 491.04 WECo No. D-164868 1.8 10 18 1000 1.8 (C) Range Calibrator TS»293/CPA-5 Ose for radar IFF. P/0 Radar Sets AN/CPX-1 and AN/CPX-2 186.3 Belmont Drawing No. A-8K-3577 1.8 10 1000 1000 1.8 Circuit Data for Figure 1-184. F in kc. R in kilohms. C in mil. L in Mb. U) (Bl (c) Figure 1-184. ModiHcotione of two-tlage feed-baek eaci/fotor to improve tine-wave output Fig. Equipment Purpose F, CR IB Rs Rs Rr Rs Rs Rs R. Rio (A) Crystal Impedance Meter TS-330/ TSM Substi- tution circuit for meas- uring param- eters of crys- tal unit 1-15 Mili- tary Stan- dard quarts sr- units 2 2 each 22 1 0.27 26 1 1 1 Circuit Data tor Figure 1-185. F in me. R in kilohma. C in accept where otherwise noted. L in fih. WADC TR 56-156 282 Saclien I Crystal Oscillators R. Rt c. C, Ci C« C» Cs Ct C, c. L, mi 2.2 2.2 3800 100,000 ■ 390 10,000 76 m CO ■ 600 6SN7GT 2.2 2.2 8300 100,000 3900 200 10,000 75 10 3900 ■ 380 6SN7GT 1 CO 6000^ 60,000 SSO 76 ■ 1 26 00 25 SigC Stock No. 2C-638- ICl 6SN7GT ffguM I-I85. ColpMts circuits modMor/ tor sortos-modo crystal control Cl C] c. c« C, Ct Ct Cl C, Li 1 L: 1 La a V| 7 . 8 - 61 1 . 6 - 260,000 260,000 10,000 80 6-42 250,000 2 . 6 - H|| |■||||| 6 V 6 GT ■■i 140 7 386 Mch each 1 1 1 1 WADC TK 56-1 M 0^ 0m &600 Equipment Purpose Crystal Impedance Meter TS« 683/TSM Substi' tution circuit for meas' uring param- eters of crystal unit Crystal Impedance Meter TS- 683/TSM Substi- tution circuit for meas- uring param- eters of crys- tal unit Test Set TS-250/ APN Range osc. Output consists of pos- itive range pips Diversity Receiving Equipment AN/FRR-3 BFO with sfc; manual or crystal opera- tion CR R. Mili- tary Stan- dard quartz crystal units 0.1 each Mili- tary Stan- dard quartz crystal units 0.0 56 each Bliley No. 122- 5006 (octal base) 100 1 1 R. R. I R. I R, I R. I R. 15 25 15 25 2.2 10 20 5 0.035 500 1.6 Circuit Data for Fig^ure 1-185. F in kc. R in kilohms. C in except where otherwise noted. L in <ih. WADC TR 56-156 284 WADC TR 56-156 285 Section I Crystal Osdilators (B) and (C) are substantially of the same basic design as circuit (A). Except for resistors R« and capacitances C3 and C4, the parameters of circuits (B) and (C) have the same numbers as their functional analogues in circuit (A). Circuits (B) and (C) are not designed for parallel-resonance measurements. For crystal resistance measure- ments, the calibrating resistor must be substituted externally for the crystal unit. The capacitors Ci are fixed and the inductors L, are continuously variable and are so ganged as always to be equal. In each of the Cl-meter circuits shown, it can be seen that if the resistance of the tank, including the crystal, were zero, and if the tank were per- fectly balanced, no voltage would exist between the crystal and ground. The voltage across Lia plus that across C,a would equal zero, and no cur- rent would flow through the resistors Ri, which effectively form a bridge between the inductance arm to the grounded connection of the capacitance arm. In practice, a net voltage does exist across L,a and CiA in series, and this voltage appears across Ria, being measurable at the jacks J1 and J2 in circuit (A). The r-f voltage across Rib is approximately that across J3 and J4, which in turn is equal to the Ria voltage plus that across the crystal resistance. The Ri resistors are not essential insofar as maintaining oscillations is con- cerned, but they load the circuit, thereby reducing the effect of the variations in crystal resistance upon the oscillator activity, and they serve to pro- tect the crystal, to balance the circuit to ground, and to facilitate measurements of the crystal volt- age (Eiie = Ej 3 — Eji) without unduly interfering with the effective circuit parameters. The Cl-meter oscillator can be analyzed as a particular type of transformer-coupled oscillator, as an impedance- inverting oscillator, or as an equivalent Pierce oscillator having a crystal X,. =: a>(L,A + L,n) and an effective crystal resistance accounting for the losses in the resistances Ri as well as in the R, of the actual crystal. 1-437. Figures 1-185 (D) and (E) are examples of grounded-plate Colpitts circuits which have been modified for series-mode crystal control. Cir- cuit (D) is designed to provide positive range pips to the grid of Vj. The circuit operates class C at either one of two frequencies, the appropriate crystal being connected between the cathode tank and the grid of V,. Circuit (E) is designed for either manual or crystal control. During manual control the resistor R, replaces the crystal unit. V, is operated as a reactance tube. The a-f-c bias varies in such a way that the b-f-o frequency tends to follow any changes in the frequency of the teletype signal being received. CRYSTAL CALIBRATION 1-438. The design of a crystal oscillator to be used for calibrating the frequency of other oscil- lators generally is directed toward obtaining out- puts rich in harmonics. Where tuned-plate circuits are required the L/C ratios should be high, so that high impedances are also presented to the overtone frequencies. The oscillator should be operated class C, and often the gridleak resistance is a megohm, or higher. If the crystal calibrator is to serve as a frequency standard of greater- than-average precision, this precision becomes the principal design problem insofar as the oscillator is concerned; if need be, the required harmonics can be developed in nonlinear amplifier stages that follow the oscillator stage. The higher the over- tone, the weaker will be its effective output power, but with proper design useful outputs up to and above the 100th harmonic can be obtained. With the addition of frequency multiplier and/or di- vider circuits a single crystal can provide a useful calibrator frequency range as broad as desired. For maximum precision, a G element, usually cut for 100 kc, should be used. 1-439. Figure 1-186 illustrates a simply designed crystal calibrator employing an electron-coupled Miller oscillator operating into a resistive plate load. Such a circuit will ensure sufficient frequency stability for most purposes. Harmonic outputs in steps of 100 kc are provided up to frequencies of 10,000 kc. For higher frequencies, the 1000-kc crystal can be used to provide calibration points in multiples of 1000 kc. The variable grid capaci- tor is employed to ensure that the crystal operates into the correct load capacitance. Crystal Calibrator Employing Regenerative Freguency Divider 1-440. Figure 1-187 shows the regenerative fre- quency-divider circuit of the crystal frequency indicator (CFI) used in Radio Transmitting Set AN/ART-13A. This circuit employs a 200-kc crys- tal to control a rich mixture of harmonics, pro- viding useful check points spaced as close as 25 kc apart. The crystal oscillator, utilizing the triode section, V„ seems best described as a modification of an impedance-inverting Pierce circuit. When oscillations first start, the output of the oscillator is fed to grid No. 1 of the pentagrid mixer, V3. The 50-kc and 150-kc components of the noise volt- ages that are mixed with the 200-kc signal are amplified by V3 and fed to the input of the Vj triode section. The V, plate circuit, which is tuned WADC TR 56-156 286 Section I Crystal Oscillators I lOOKe 5 iOOO S I'SMEO S I00> X Kc > >2000 > <0HM8 * eooo- • **9® Ngun I- 166. Typico/ des^n of sinplo-tvho gonorol-purposo crystal calibrator circuit Fa«Fl 'F2-S0KC aOOKCag^B^ Rl f If •OOK 30 T C7 04 Mf Whsskc N0TE = UNLESS otherwise SHOWN ALL CAPACITORS ARE IN MtCROMiCROFARAOS ALL RESISTORS ARE IN OHMS. SPEECH AMPL INPUT figure I-I87. CFf regefierative froquoncy divider In Radio Transmlttor T-47A/AftT*l3 (P/O Radio Transmitting Sot AN/ART-13AJ WADC TR 56-156 Section I Crystal Oscillators to 150 kc, amplifies the 150-kc noise input and triples the 50-kc input. The 150-kc output of Va is then fed back to the pentagrid mixer at grid No. 3. It is again amplified and fed back to Va. How- ever, the direct amplification and regeneration of the 150-kc signal alone is not sufficient nor prop- erly phased to maintain oscillations at this fre~ quency. The 150-kc oscillations are sustained prin- cipally by tripling the 50-kc feedback, which builds up as the amplified difference frequency of the 200-kc and 150-kc inputs to Vs. The output of Vs is effectively a 50-kc fundamental frequency stand- ard of large harmonic content that is fed to the grid of triode section V 4 , where it is mixed with signals from the variable oscillators of the trans- mitter. The output of V 4 is fed to the input of an audio amplifier, which amplifies the beat note whenever the variable oscillator approaches the frequency of one of the CFI harmonics. In prac- tice, the recommended check-point harmonics are spaced 25 kc apart in the 200— ^00-kc frequency range, 100 kc apart from 2000 to 3000 kc, 150 kc (3000—4000 kc), 200 kc (4000—6000 kc), 300 kc (6000—9000 kc), 450 kc (9000—12,000 kc), and 600 kc (12,000 to 18,100 kc). The presence of the harmonics of an apparent 25-kc fundamental, which is used in the low-frequency calibrations, is not readily explained on the basis of the foregoing discussion of the circuit. A complete analysis of the nonlinear characteristics of the circuit is not available, but it appears possible that if a 25-kc signal appears at the plate of Va, it can conceiv- ably be sustained by being fed to V^, mixed vnth f. to form a sum frequency of 175 kc, fed back to Vn and mixed with the 200 -kc injector signal to regenerate a difference frequency of 25 kc. It 3 should be understood that f^ — 3f.s =-i-fi is a nec- 4 essary relation, and that f^ and fa are synchro- nized and controlled by the crystal oscillator. The phase and frequencies of the regenerative circuits automatically follow the phase and frequency of the Vi output. For a more analytical study of re- generative frequency dividers, see discussions by R. L. Miller, R. L. Fortescue, and W. A. Edson. SYNTHESIZING CIRCUITS 1-441. Of great promise, particularly for use in airborne radio equipment in the v-h-f range where crystal control is necessary to maintain the re- quired frequency stability, has been the develop- ment of synthesizing circuits, in which a very few crystals are able to control a large number of channels. In the discussions to follow we shall use the term frequency synthesis very loosely to apply to any type of frequency-control circuit or system in which a few fixed-frequency oscillators are used to control or to stabilize a large number of radio frequencies. If the term were used rigor- ously, it would apply only to those cases where an output frequency is produced entirely from heter- odyned combinations of internally generated fre- quencies. Examples of this type are provided by the Plessey frequency generator and by the Collins transmitter frequency-control system employed in Radio Set AN/ARC-27. For our purposes we shall extend the term to cover such systems as the Bendix frequency-control circuit in Radio Set AN/ ARC-33, where the output frequency is not actually synthesized but is obtained from a vari- able-tuned master oscillator that is crystal-stabi- lized at many frequencies. Also implied by the term will be such systems as the Collins crystal- controlled multichannel receiver circuits. In these latter circuits only one end-product frequency is desired — a fixed superheterodyne intermediate frequency. But the system design is such that with the use of a very few crystals the desired inter- mediate frequency can be synthesized under crys- tal control from received signals on any one of hundreds of possible radio channels, t The Plessey Synthesisiiig System* 1-442. The first crystal-controlled frequency syn- thesizer in commercial usage appears to have been f Not all types of synthesizers in current use are covered in the above discussion. Other recently developed and equally important circuits include: The General Radio Company synthesizer, developed under SiCTal Corps Contract No. DA-86-039-SC-16642. The GR synthesizer operates on a principle fundamentally different from those described in this report. In the GR system an oscillatoi' is ^ase- and frequency-locked through a variable scale-of-N divider. The pulse output of the divider is compared by coincidence methods with a pulse derived from a crystal oscillator. The frequency range of this synthesizer is 0.1 me to 10 me. The Matawan Synthesizer ME-447, of the Lavoie Laboratories Insti'ument Company. This system generates any multiple of 1 kc within the range of 1.0 me to 2.0 me. The Rohde and Schwarz decade synthesizer and exciter system (Federal Telephone and Radio Company Types HS-431, HS-441, and HS-471), which covers a range of 50 kc to 30 me. The Telefunken Precision Frequency Meter. This meter is used in the measurement of frequencies between 1 kc and 300 me. The circuitry contains a frequency synthesizer capable of generating sine-wave outputs between 1 kc and 30 me. It is claimed that harmonics and sidebands of the output frequency are at least 80 db below the selected signal, and that the syntheslizer accuracy is ± 0.2 cps for frequencies between 1 kc and 3 me, and ± 2.0 cps between 3 me and 30 me. * Note: The discussion of the Plessey synthesizing system is based primarily upon the report, “The Frequency Synthesizer”, by Mr. H. J. Finden of the Plessey Com- pany, Ltd., England, published in the Journal of the Inutitution of Electrical Engineers, Vol. 90, Part III, 1943. WADC TR 56-156 288 that developed by the Plessey Company, Ltd. of England. This synthesizer has been designed as a frequency generator to be used in making precise radio-frequency measurements. The synthesizing system employed is nevertheless quite applicable for other uses, such as providing multichannel ex- citation voltages for radio communication equip- ment. As designed by the Plesaey engineers, the synthesizer generates a sequence of harmonic sig- nals of much greater precision and purity than is obtainable with conventional tjrpes of frequency generators. The original model permits a direct- reading dial selection of any of the first 10,000 harmonics of 1 kilocycle per second; a later and larger model extends the range to the first 100,000 harmonics, i.e., any harmonic of 1 kc up to 100 me. Ail these frequencies are made available singly as pure sine waves (unmixed with other har- monics or frequency products) by a decade system cf frequency dividers and multipliers, mixing stages, and filters where all the generated fre- quencies are under the control of a single 1000-kc precision crystal standard. Theoretically the sys- tem could be extended to cover a broader or a dif- ferent frequency range; or could be changed to permit steps between adjacent frequencies that are smaller or larger than 1 kc. If required, it would be quite practicable for the Plessey gener- ator, itself, to be expanded to cover also the 100- to-lOOO-mc range in 10-mc steps. In 1965 the Schomandt Company of Munich, Germany placed on the market a similar type of synthesizer fre- quency generator covering ^e 0 — 30-mc range in 1-kc steps. The output of the Schomandt synthe- sizer is equivalent to that of the Plessey synthe- sizer in quality, having at least a 60-db attenua- tion of all unwanted frequencies. It was the demand for such narrowly spaced pure output fre- quencies for use in making frequency measure- ments that originally led to the development of the synthesizer circuits. SYNTHESIZER ADVANTAGES IN RADIO- FREQUENCY MEASUREMENTS 1-443. Prior to the development of the frequency synthesizer there were two conventional methods for measuring radio frequencies — the “interpola- tion” method and the “successive heterodyning" method.* Briefly, the interpolation method, which is satisfactory where extreme accuracy is not re- quired, consists of mixing the unknown frequency with the two nearest harmonics of a frequency standard, and zero-beating the difference frequen- cies obtained against the output of a linearly tuned variable oscillator. It is then possible to interpolate the unknown frequency by determin- Section I Crystal 'Oscillalort ing its relative position between the known har- monics of the standard. In the successive hetero- dyning method, the unknown frequency is mixed with a known harmonic of a frequency standard ; the difference frequency is then heterodyned with a second standard harmonic to obtain a second and lower difference frequency; and the process is repeated, if necessary, until a difference fre- quency is obtained that lies within an accurately measureable audio range. Although the successive heterodyning method can be quite accurate, occa- sions arise where the operator cannot be certain without undue checking that the difference fre- quencies being measured are not the products of unwanted harmonics contained in the heterodyned signals. The use of a frequency synthesizer that permits individual pure sine-wave outputs of a sequence of narrowly spaced frequencies, instead of a simultaneous mixture of many harmonics, can be said to offer a third and greatly superior means of measuring radio frequencies. 1-444. With the use of decade dial control, greater operating simplicity is possible than with the in- terpolation method ; and when the pure sine-wave frequencies are spaced only 1 kc apart, the inter- polation accuracy of the successive heterodyning method is maintained, but with the elimination of those chance difference products that can result from harmonic mixtures of multiple stages of heterodyning. FUNCTIONAL OPERATION OF PLESSEY SYNTHESIZER 1-445. The circuit system by which the Plessey synthesizer produces thousands of frequencies, all controlled by a single 1000-kc crystal standard, is illustrated in figure 1-188. The block diagram shown is that of the original, single-cabinet model that permits the operator a choice of any one of the first 10,000 harmonics of 1 kc. It can be seen that there are three successive stages in which the input frequency is divided by 10, so that the last divider represents an over-all division of the orig- inal standard (1000 kc) by 1000. The dividers and the 1000-kc harmonic generator are of the syn- chronized, free-running, multivibrator type whose outputs are rich in harmonics. Each of these mul- tivibrator circuits forms the first stage of a sequence which can be tuned to pass any one of the first 10 harmonics of its respective multivi- brator fundamental. These sequences are labeled A, B, C, and D in figure 1-188. In the synthesis of a frequency, we can say generally that sequence A * Note: See paragraph 2-66 to 2-151 for detailed descrip- tions of frequency'-nreasuring systems in current use. WADC TR 56-156 289 SYNTH E S I Z ING C I R C U I T S Section I Crystol Oscillators A CIRCUITS B CIRCUITS C CIRCUITS D CIRCUITS Figuro 1-188. Blotk diagram of a P/ossoy tyntho$iior dotignod to cover the 0 — 10-mc spottrum In 1-kc steps supplies that part of the final frequency which is a multiple of 1000 kc, B that part which is a mul- tiple of 100 kc, C that part which is a multiple of 10 kc, and D that part which is a multiple of 1 kc. 1-446. For example, assume that an output fre- quency of 6789 kc is desired. The A, B, C, and D harmonic selectors, respectively, will be decade- set to pass the 6th, 7th, 8th, and 9th harmonics of their respective input signals from the preced- ing multivibrator stages. In balanced modulator C, the output of selector D, 9 kc, is mixed with the 80-kc output of selector C. (The signals are heterodyned in a balanced modulator circuit rather than in « more efficient type of mixer in order to eliminate the two input frequencies from the modulator output. In this manner the sum and difference products become the dominant frequen- cies in the modulator output.) Filter C is dial-set to pass the desired frequency product, 89 kc, which it feeds to balanced modulator B. In modu- lator B, the 89-kc signal is heterodyned with the 700-kc output of the decade-set harmonic selector B. Filter B is dial-set to pass the sum product, 789 kc, from the B modulator output to the A modu- lator input, where it is mixed with the 600()-kc output of harmonic selector A. Filter A is dial-set to pass the sum product, 6789 kc, of the mixed signals, which product is then amplified and fed through a phase inverter to the synthesizer out- put jack. 1-447. The foregoing example of the operation of the Plessey sjmthesizer suggests that the sum rather than the difference products of the mixed signals are always selected. In practice this is not the case, even though the decade dialing system is so designed that the operator is always provided a direct reading of the output frequency as if he were only adding the decade units together. In order to sufficiently filter out the unwanted prod- uct, it is important that the signals to be mixed are so selected that there is at least a 10 per cent difference in frequency between the sum and dif- ference products. Since the filters must be capable of suppressing ail adjacent harmonics of the mixed signals, it can be assumed that they are also cap- able of suppressing the unwanted heterodyne product if it differs from the desired product by as much as the fundamental harmonic of the mpd- ulator input from the harmonic selector. For ex- ample, in modulator C, the space between the sum (fc -b fi>) and the difference (fc — fu) frequen- cies should not be less than 10 kc, the fundamental of the harmonic from selector C. Since (fc -f- f,.) — (fc — fi.) = 2f„ > 10 kc then fr, must never be less than 6 kc if it is to be WADC TR 56-156 290 Section I Crystal Oscillators mixed with fc. Similarly, fd, must not be less than 50 kc if it is to be mixed with fs, and fnci> must not be less than 500 kc if it is to be mixed with fA. 1-448. To illustrate, let us suppose that a fre- quency of 91 kc is desired. It would not do for fc and fi) to be 90 kc and 1 kc, respectively, for then the sum product, 91 kc, would be separated from the difference product, 89 kc, by only 2 kc. Rather, 100 kc should be selected as fc and 9 kc as fa. The variable filter C would be set to pass the difference product, 91 kc; which product differs from the sum product, 109 kc, by 18 kc, well beyond the minimum permissible limit of 10 kc. 1-449. As a more involved example we shall de- termine the heterodyne frequencies that would be used in the synthesis of an 8136-kc output. For a mental calculation of the correct frequency com- binations the easiest method is to start with the output frequency, fABcu, and from this determine fA, fnei), fa, fcii, fc, and f„ in that order, working from the larger units to the smaller. Each of the above six frequencies is determined by remember- ing that none of the input frequencies to the A, B, and C modulators can be less than 500, 50, and 5 kc, respectively. Thus, we see at once that 8136 kc is not to be the sum product of 8000 kc and 136 kc in the A modulator, since 136 kc is less than 500 kc. So f^ must be 9000 kc and f„ci) must be 1000 minus 136 kc, that is, 864 kc; which means that filter A will be adjusted to pass the difference product (9000 kc minus 864 kc). Since 64 kc is greater than 50 kc, the required 864-kc output of modulator B can be obtained as the sum product of 800 kc and 64 kc, fn and fen, respectively. Since 4 kc is less than 5 kc, the required 64-kc output of modulator C must be obtained as the difference product of 70 kc and 6 kc, fc and f,,, respectively. We see that in order to select an output of 8136 kc, the decade dials of the A, B, C, and D har- monic selectors must be set to pass, respectively, the 9th, 8th, 7th, and 6th harmonics. In other words, the output frequency would be a synthetic product of the four frequencies, 9000 kc, 800 kc, 70 kc, and 6 kc. So also would be an output fre- quency of 9876 kc. Since the decade dials that con- trol the harmonic selectors may be set at the same positions for two or more frequencies, some ar- rangement must be made so that the decade read- ing presented to the operator identifies correctly the particular frequency being synthesized. This convenience is accomplished in the Plessey syn- thesizer by manually operated range adjustments that alter the correspondence of the dial readings with the dial positions. Thus, in the example above, with the proper range settings, decade dial A in position 9 would give a reading of 8, decade dial B in position 8 would give a reading of 1, decade dial C in position 7 would give a reading of 3, and decade dial D in position 6 would give a reading of 6. The mechanics of exactly how this feature is incorporated in the Plessey syn- thesizer, although relatively simple in principle, is somewhat beyond the subject matter of our assignment here. CIRCUIT DESIGN OF PLESSEY SYNTHESIZER 1-450. The general circuit design employed in a Plessey synthesizer is shown in the schematic dia- gram of figure 1-190. The circuit shown, when synchronized by a 1000-kc standard (whose cir- cuit is not shown), is capable of covering the 0 — 10-mc range in 1-kc steps. Note that each of the four decade harmonic sequences begins with a multivibrator-type of harmonic generator. Rheo- stats are furnished for adjusting the natural oscil- lation period of each multivibrator, in order to allow for aging effects and the like. More elab- orate or reliable harmonic-generator circuits are not required since the failure of any of the multi- vibrators would be immediately apparent by the reading in the output meter. Figure 1-189 shows in detail the circuit parameters of the 100-kc mul- tivibrator, which also acts as the 1st divider. Note that the 1st divider is synchronized by the output of the 1000-kc amplifier and not directly by the frequency standard. The output of the 1st divider in turn is used to synchronize the 2nd divider, and that of the 2nd to synchronize the 3rd. 1-451. In figure 1-190 it can be seen that harmonic selection is achieved by switching to the correct tuning capacitor from a bank of 10. The same inductance is used for each of the harmonics. Since the percentage difference between adjacent harmonics is less as the order of the harmonic be- comes higher, it is more difficult to eliminate the 9th and 11th harmonics when selecting the 10th,. than it is to eliminate the 1st and 3rd when select- ing the 2nd. For this reason, the value of each of the fixed tuning inductors is chosen to provide an optimum Q at the 10th harmonic. This permits a relative magnification of the 10th harmonic over its adjacent harmonics of approximately 200, which is equivalent to a 32-db attenuation of the 9th and 11th harmonics and more than that for all others. The attenuation of adjacent harmonics be- comes greater as the selected harmonic becomes lower, so that in any event it is never less than 32 db at each tuned circuit. Two tuned circuits in series provide more than a 60-db attenuation. WADC TR 56-156 391 Section I Crystal Oscillators Figure 1-189. Schematic diagram of a Plessey syn- thesizer designed to cover the 0 — lO-mc spectrum in I -he steps. The circuit of the crystal oscillator standard is not shown which for all practical purposes is sufficient to consider the selected harmonic a pure sine wave. 1-452. With the use of a balanced modulator it is not necessary to use as many tuned circuits as would otherwise be necessary to eliminate all un- wanted harmonics and frequency products. Note in figure 1-190 that the balanced modulator design is such that two matched amplifiers have a com- mon output circuit, but that they are excited by equal signals 180 degrees out of phase, so that the amplified signals cancel each other in the load. Thus, even though the heterodyne efficiency of the balanced modulator is less than that of other types of mixers, the balanced circuit is greatly advan- tageous in helping to eliminate all the unwanted frequencies, particularly the unwanted harmonics, that originate in the circuits preceding a mixer stag!'. In the Plessey synthesizer it can be seen that the modulators are provided with a switching arrangement by which one of the tubes of each modulator can be cut out of the circuit by opening its cathode return. One of these switches is opened whene\ or a modulator stage must pass an un- mixed signal. With one tube removed, the balanced arrangement is destroyed, and since only one in- put signal is being handled, the vacuum tube still connected in the circuit will be operated as a con- ventional amplifier. If, for example, the desired output were a 2000-kc signal, none of the modu- lators would be in operation except modulator A, which would be unbalanced and operated simply as an amplifier of the 2nd harmonic from the 1000-kc harmonic generator. 1-453. Variable-tuned circuits are provided as bandpass filters. These must be adjusted manually in selecting the proper heterod 3 me product to be passed. The selectivity is sufficient to provide at least a 30-db attenuation of any unwanted signal that differs as much as 10 per cent from the de- sired signal. 1-454. The phase inverters are inserted for proper impedance matching. They permit an output at any frequency within the operating range of 100 millivolts across a 75-ohm load. The system as a whole insures at least a 60-db attenuation of all unwanted frequencies. The Beadix Synthesisiiig System 1-466. In America, much of the pioneering in the field of frequency synthesis has been done by the research staff of the Bendix Corporation. The following discussion is based upon the syn- thesizing circuit originally described by W. R. Hedeman of Bendix in the magazine Electronics. Figure 1-191 shows a block diagram of the syn- thesizer circuit developed At Bendix for use in controlling the frequency of a continuously vari- able v-h-f receiver heterodyne oscillator. In this circuit, the first crystal oscillator employs but one crystal. The harmonic generator that follows this oscillator produces a rich output of harmonics, the first of which is f^, the fundamental of the first crystal oscillator. The harmonics selector is com- posed of a number of band-pass circuits, each cir- cuit designed to pass a particular harmonic of the crystal frequency. The number of frequencies con- trolled by the synthesizer is directly proportional to the number of harmonic channels in the selec- tor. Let fh equal the harmonic selected and f„ equal the frequency of the variable oscillator. The value of f„ is always higher than that of fh. These two frequencies are mixed in the first frequency con- verter to form the sum-and-difference frequencies, which, in turn, are fed to the input of the first band-pass amplifier. The first band-pass amplifier amplifies and passes only the difference frequency, f„ — fi,. This difference frequency is fed to the second frequency converter, where it is mixed with the output, f„ of the second crystal oscillator. The second crystal oscillator is generally provided with more than one crystal unit, but only the funda- mental frequency of the oscillator is used when a WADC TR 56-156 292 WADC TR 56-156 Section Crystal Oscillatoi D CtRCUITS Figure 1-190. Schematic diagram at the 100-kc synehroniied multivibrator used in thu Piessey.synfhesizer at a ^ decade divider of a 1000-kc $tandard and at a lOO-kc harmonic generator n 293-294 SccHon I Crystal Oscilloters Figure 1-191. Block diagram of froquoney-tynthotixor circuit particular crystal is selected. The number of con- trolled channels is directly proportional to the number of second-oscillator crystals. The sum and difference frequencies of the frequency converter are fed to the second band-pass amplifier, which amplifies and passes only the difference frequency, <f„ — fh) — f.. This difference frequency is fed to a discriminator. The number of channels con- trolled is directly proportional to the number of discriminators used. The d-c a-f-c output of the discriminator is used to control the bias of the control tube. The plate current of the control tube determines the rotor position of a servo motor, and the rotor is mechanically coupled to control the tuning: elements of the variable oscillator. The servo motor continues to turn and thereby con- tinues to change the frequency, f„, until the output of the discriininator is zero. This occurs when the output of the second band-pass amplifier is equal to frt, the frequency of the discriminator circuit. By reversing the polarity of the discriminator out- put leads, f„ can be made to vary in the opposite direction in order to reach equilibrium. Thus, for each value of fh, f„ and fd, there are two equili- brium values of f„. These are given by the equation f„ = fh + f. ± fd 1^66 (1) 1-466. Let N = Total number of channels, f„. H = Number of harmonics used (1st crystal oscillator) . X = Number of crystals (2nd crystal oscillator). D = Number of discriminators. It can be seen from equation 1 — 456 (1) that N = 2HXD 1—456 (1) The factor 2 is introduced by the fact that for each discriminator there are two values of f„ for each combination of fh and f,. One value is f„ = fh + f, + fd, and the other is f„ = fh + f, — fd. 1-457. As a concrete example, let us imagine that it is desired to cover the frequency range between 100 and 156 me with the channels spaced 200 kc apart. The lowest value of f„ is to be 100.2 me, and the highest is to be 166 me. Thus, N = — (min) f« ^ ^ _ 156 — 100 _ Afo 0.2 1—467 (1) By equation 1 — 456 (1), HXD = = 140 1—467 (2) The smallest total number of elements occurs when H, X, and D can be made as nearly equal to each other as possible, but in an actual design problem, this may not be the most practical solution. In our particular example let us assume that the first seven harmonics of L are to be used. With H = 7, then by equation (2) XD = .3^ = 20 1-457 (8) WADC TR 56-156 295 section I Crystal Oscillators The combinations (X,D) possible are (20,1), (10,2), (5,4), (4,5), (2,10), and (1,20). For our problem we shall suppose that the combination (X = 10, D = 2) proves the most practical. Thus, with the use of 11 crystals in all and 2 discrim- inators, 280 crystal-controlled channels are to be obtained. 1-458. Let ft, fi', f;i . . . . fjsti designate the values of f„ from the lowest to the highest, in that order. Let fxt, f,;. .... fxtn designate the values f, from the lowest to the highest, in that order. The values of f„ in ascending order are f,, 2f, .... 7f,.. Fi- nally, let frt, and f.i-.. designate the lower and the higher discriminator frequencies, respectively. To avoid the possibility of spurious conversion fre- quencies, the highest value of f,, should be less than the minimum associated 1st band-pass amplifier frequency, f„ — f„. Also, the lowest value of f, should be higher than the highest 2nd band-pass amplifier frequency, which, of course, will equal the highest discriminator frequency, fd. The order of the variable-oscillator frequencies and the se- quence of circuit connections required to provide each frequency is indicated by the following se- quence of equations. li + fxl — fd 2 100.2 mc 11 + fgl — fdi = 100.4 mc 11 + fx. — fd 2 = 100.6 mc fd =f,. + fx. — fd, = 100.8 mc etc f^., - t fxHI — fd, = 104 mc L, = f. + fxl -b fdi = 104.2 mc f-' = f. + fxl 4 - fd-. = 104.4 mc f..:, = f,. + fxj 4- fdi = 104.6 mc f^d = f.. + fx-J 4- fd 2 = 104.8 mc etc fd„ - f, + fxio 4- fdj = 108 mc fdi - 2 f,. + f„ — fda = 108.2 me fd - 2 f,. 4 - L, — fd, = 108.4 me f,:! = 2 f.. -h f,a — fda = 108.6 mc etc f-sn = 7 f,. -f f.m + frtj = 156 mc 1-459. It can be seen from the equation sequence in paragraph 1-458 that the difference between the channel frequencies is equal to the difference between the discriminator frequencies. Thus, Afd = fd 2 — Li = aL = fj — fi = 0.2 mc 1—459 (1) Note in the equation sequence in paragraph 1-458, that for a given harmonic frequency, L, all the f, crystals are used in sequence before the polarity of the discriminator outputs are reversed. In other words, fdi and fdo are first subtracted from all com- binations of a particular harmonic with the X- crystal frequencies and then added to the same combinations. This process is repeated with each harmonic. If we subtract the equatibn for f| from the equation for L, we have Afx = f ,2 — f,i = L — fj = 0,4 mc = 2 Afd 1—459 (2) In the general case, Afx = D Afd 1—459 (3) where D is the number of discriminators. The same value of Af, also holds between any other two consecutive values of fx. Thus, fx2 = fxl + Afx f.:i = f„ + 2 Afx f,n = fx. + (n - 1) Af, = fx, + (n — 1) D Afd 1—459 (4) The highest frequency of the second crystal oscil- lator is given by equation (4) when n = X = the total number of crystals. Thus, (max) fx = fxl + (X — 1) D Afd 1_459 (5) With the use of equation (6) we can find the low- est discriminator frequency, fdi. This is done by subtracting the equation for fi>,i from the equation for fj,, which gives f«i — fat, = fxl — fxKi 4" 2fdi = Afd =- 0.2 mc 1—459 (6) Since f„„ is a particular case of (max) f,, we can substitute equation (5) in equation (6) to obtain a general equation. We find fx. — f.i - (X - 1) D Afd -t 2 fd, = Afd WAOC TR 56-156 296 Section I Crystal Oscillators On rearranging after canceling out f.i , For the fundamental of the 1st crystal oscillator, fdi = Afa ~ 1—459 (7) The general equation for any particular discrim- inator frequency, fa„, is similar to that for f,„ given by equation (4) . Thus, , fan = fa, + (n - 1) Afa. 1—459 (8) and (max) fa = fa, + (D - 1) Afa 1—459 (9) The next problem is to obtain a general equation for fc. This can be had by subtracting the equa- tion for fao from the equation for fa,. The re- mainder is Afa = fo + f,i — f,io — 2 fa- where f,,o = (max) f, and faa = (max) fa. Thus, f. = 2 fas — f,i -I- f.i + (X — 1) D Afa + Afa or f„ = 2 fa- + 2 fa. fc = 2 [(max) fa + (min) fa] 1 — 459 (10) Finally, with fc determined, we can use the equa- tion for f, tofindfxi. 1-450. We are now in a position to express any of the circuit frequencies in terms of the param- eters fi, Af„, N, H, X, and D. For the nth channel, f„ = f, + (n — 1) Af„ 1—460 (1) For the highest channel, (max) f„ = f, + (N — 1) Af„ 1—460 (2) For the lowest discriminator frequency, fa, = D) 1_460 (3) For the nth discriminator frequency, fa„ = At (DX — D + 2n — 1) For the highest discriminator frequency, At (DX + D — 1) t = 2 At D X = 1—460 (6) For the nth harmonic frequency, t,. = 2 At D X n 1—460 (7) For the highest harmonic, (max) t — At N 1 — 460 (8) For the lowest frequency of the 2nd crystal oscil- lator, t. = 2f,-At (3DX-D + 1) For the nth frequency of the 2nd crystal oscillator, „ _ 2 f, — At (3 DX -f D + 1 — 2 D„) Ixn 2 1—460 (10) For the highest frequency of the 2nd crystal os- cillator. (max) t = 2f, — At (DX + D + 1) 1—460 (11) and the difference between the consecutive values of t. At - Af„ D 1—460 (12) (max) t 1—460 (5) 1-461. On applying the equations in paragraph 1-460 to the numerical example that has been assumed, where f, - 100.2 me, Af, - 0.2 me, N = 280, H = 7, X - 10, and D = 2, we find that . 0 . 2(1 + 20 — 2 ) ti = 2 tj = 1.9 + 0.2 = 2.1 me . 0.2 X 280 . I,. = — 8 me fi,i. fill’, etc. - 8, 16, 24, 32, 40, 48, 56 me f,^g»M^g |0 . -2 + l) ..04.3 me At = 0.4 me ti.t-, etc = 94.3, 94.7, 95.1, 95.5, 95.9, 96.3, 96.7, 97.1, 97.5, 97.9 me WADC TR 56-156 297 Section I Crystal Oscillators Note that the highest harmonic frequency is equal to the bandwidth of the frequency range being covered. 1-462. From the equation, f„ = fh + f, ± fd, it can be seen that the frequency stability will be approximately the stability of the crystal oscilla- tors. This is because f^ is so very much lower than fh and f,. Although the discriminators should be designed with low-loss, temperature-compensating materials, even a large percentage variation in f^ would be negligible in its percentage effect upon f„. Since it is the sum of fi, and f, that determines fo, the maximum percentage frequency deviation of the total can be no greater than that of the crystal oscillators individually. Without oven con- trol, the channel frequencies can be maintained within a tolerance of ±0.005 per cent, and better. RADIO SET AN/ARC-33 1-463. Radio Set AN/ARC-33 is an airborne re- ceiver-transmitter designed to operate in the v-h-f and u-h-f spectrum. This equipment, developed by the Bendix Corporation, employs a modified ver- sion of the Bendix frequency-synthesizing system discussed in the foregoing paragraphs. The fre- quency-control section (see figure 1-192 for block diagram) is designed to permit receive-transmit communication on any one of 1750 channels spaced 100 kc apart in the 226-to-399.9-mc band. 1-464. An important modification in the synthe- sizing system arises from the fact that the syn- thesized frequencies are utilized as heterodyne in- jection signals during reception and as carrier signals during transmission. The tuning controls are such that the receiver and transmitter circuits are always automatically tuned to the same chan- nel, but since the desired receiver injection fre- quency must differ from the tuned channel fre- quency by an amount equal to the receiver inter- mediate frequency, the design engineers had to decide whether to let the variable-frequency oscil- lator be, in effect, a subharmonic local oscillator for the receiver or a subharmonic master oscilla- tor for the transmitter. They decided in favor of the receiver. Thus, the stabilized output frequency, f„, of the vfo, after being multiplied 12 times, is used directly as the injection voltage in the 1st mixer stage in the receiver. This provides a fixed intermediate frequency, f,, of 15.325 me for each of 1750 channels. Now, the v-f-o output, fo, is also used in the synthesis of the channel frequency of the transmitter. Since the 12th harmonic of f„ always differs from the channel frequency by an amount equal to f,, all that needs to be done in principle is to mix 12f„ with a fixed oscillator fre- quency equal to fi and for the sum product to be isolated and amplified for use as the transmitter carrier. In the ARC-33 transceiver this effect is achieved by mixing the output, f„ of a crystal- controlled oscillator (called the “sidestep” oscil- lator) with the 6th harmonic of f„, then selecting the sum frequency (6f„ plus f.) and doubling it to form the carrier frequency, f., which, after ampli- fication, is fed to the antenna. Summarizing these frequency relations in the form of equations, we have: injection frequency = 12 f^ intermediate frequency = f, = f, — 12fo antenna (channel) frequency = f, = 12f„ + fi = 2 (6fo + f.) sidestep frequency = f, = fi/2 Note the necessary harmonic relation between the sidestep output and the intermediate frequency. It is also important to note that the principle in- volved in the use of a sidestep oscillator permits, not only a Bendix synthesizing system, but any synthesizing system to be readily modified for the dual-purpose requirements of transceiver fre- quency control. 1-465. Comparison of figures 1-191 and 1-192 will reveal that the 2nd crystal oscillator in the basic Bendix synthesizer has been replaced by two crystal oscillators (the 2nd and 3rd in figure 1-192) in Radio Set AN/ARC-33. This has been done to permit a greater number of frequencies with a fewer number of crystals. 1-466. Another significant modification occurs in the ARC-33 discriminator circuit. As is explained in more detail in a subsequent paragraph, the ARC-33 discriminator is not a conventional type that employs a parallel tuned circuit to control the phase differences between the input voltage com- ponents. In that type of discriminator the output voltage always has a net d-c component unless the input frequency is equal to the antiresonant fre- quency of the tuned tank. The polarity of the d-c % component is an index of whether the input fre- quency is higher or lower than that at which the tank is tuned, and the amplitude of the d-c com- ponent can be a measure of the amount of differ- ence between the two frequencies. It is this type of discriminator that is assumed in the discussion of the basic Bendix synthesizing system, the d-c output of which is used to control the variable oscillator tuning. In the ARC-33, however, the discriminator does not employ a tuned circuit, but instead is fed a signal that is controlled by the 4th crystal oscillator. The discriminator also receives WADC TR 56-156 298 WADC TR 56-156 299 Figure 1-192. Block diagram of frequency control section of Radio Set AN/ARC-33 showing modified version of Bendix synthesizing system Saclien I Crystal Oscillators an input signal from the 4th mixer. The two sig- nals combine in the discriminator Circuit to pro- vide a net d-c output only when the two signals are of the same frequency. When the frequencies are identical, the behavior of the discriminator is quite similar to the behavior of one that employs a tuned circuit, the d-c output depending upon the differences in phase between the input voltages. Since the 4th oscillator can be controlled by either one of two crystals, this arrangement is equiva- lent to having two discriminators of the tuned- circuit type. 1-467. In paragraph 1-456 it is explained that for each discriminator, two values of f„ are possible. In Radio Set AN/ARC-33 only one of these values is used for each 4th-oscillator frequency, namely fo = fh + f. + fd 1-467 (1) In the equation above, ft, is the selected effective harmonic of the 1st crystal oscillator, equal to fm on the low band and to (fhi -f- fi, 2 ) on the high band, f, equals the sum of the frequencies of the 2nd and 3rd crystal oscillators (f ,2 + f.s), and fa is the frequency of the 4th crystal oscillator fed to the discriminator. 1-468. Other modifications of the Bendix system as occur in the frequency-control circuits of Radio Set AN/ARC-33 are of an even less radical nature than those described above. There is the division of the 1st bandpass stage into low-band and high- band circuits, and there is the addition of an a-f-c reactance tube, which is actually more of an ex- tension of the modification caused by the use of a crystal-controlled discriminator, but these and other special circuit arrangements are best ex- plained in the more detailed analyses later. At this point it will be helpful to examine briefly the role each of the various oscillator frequencies plays in controlling the final antenna frequency. 1-469. First, we shall examine the simplified block diagram shown in figure 1-193. The frequency- control system indicated represents an imaginary synthesizing circuit that provides the same trans- mitter output frequencies, f», as does Radio Set AN/ARC-33. The principal difference between the TO TUNSWTTER Figure 7-193. Simplified block diagram of the frequency-conlrol section of Radio Set AN/ARC-33 at it would appear if the variable frequency otcillator were the direct generator of the transmitter output frequency with- out the use of multiplier or side-step circuits WADC TR 56-156 300 imaginary and the actual systems is that the imaginary system is hot required to provide an injection voltage for a receiver heterodyne circuit, nor is it required to employ multiplier stages fol- lowing the variable oscillator. In other words, f„ can be assumed to equal f„. Under these conditions the various crystal and harmonic frequencies would assume the simple values shown in figure 1-193. The actual circuit frequencies are those in- dicated in the frequency diagram of figure 1-194. A comparison of the frequencies in the two sys- tems will show that the injected frequencies in each mixer stage of the imaginary system vary from one to the next in steps that are 12 times greater than are the corresponding steps in the actual system. This does not mean, except in the case of the 1st crystal oscillator and its harmonics, that the ima^nary crystal frequencies are 12 times the actual crystal frequencies ; it is only the differences between adjacent frequencies that are related in the proportion of 12 to 1. Since the fre- quency of the variable frequency oscillator in the actual circuit is eventually multiplied 12 times, it can be seen that the frequency steps in the actual circuit are equivalent to those in the imaginary circuit insofar as they add or subtract in the con- trol of the antenna frequency. Thus, we can say that the antenna frequency is effectively synthe- sized in 10-mc units by the Ist crystal oscillator and harmonic generator, 2-mc units by the 2nd crystal oscillator, 0.2-mc units by the 3rd crystal oscillator, and finally to the nearest 0.1-mc unit by the discriminator and 4th crystal oscillator. Detailed Circuit Description 1-470. The principal component of Radio Set AN/ ARC-33 is Receiver-Transmitter RT-173/ ARC-33. The receiver-transmitter is divided into a number of sectional components, two of which are of im- portance to us : a. The monitor chassis, which contains all the crystal circuits for controling the variable- frequency oscillator. b. The r-f head, which contains the variable- frequency oscillator, the multiplier circuits, the 1st i-f mixer, the sidestep oscillator, as well as the r-f amplifiers of the receiver and the power amplifiers of the transmitter. Also of importance to us are the relays which control the tuning motor. These are mounted on the main frame. We shall discuss the monitor circuits first, and then those in the r-f head. Except for occasional in- sertions and editing, the descriptions to follow are largely extracts from USAF Technical Order No. 12R2.2ARC33-2. Section i Crystal Oscillotors Monitor Chassis 1-471. The monitor chassis in Radio Set AN/ , ARC-33 concerns only the frequency control of the variable-frequency oscillator and electronic control of the tuning-capacitor drive motor with its clutches. There are no other circuits involved. The detailed circuit descriptions of the mon- itor chassis are made with reference to the com- ponent symbols employed in the block diagram of figure 1-195 and the schematic diagram of figure 1-196. With the exception of a coaxial connector for the r-f input from the variable-frequency oscillator, all external connections to the unit are made through a single connector, which is so arranged that connection automatically is made when the chassis is inserted in its proper place in the main frame. 1-472. FIRST CRYSTAL OSCILLATOR. The 1st crystal oscillator is a single-frequency, funda- mental-mode, 833.333-kc oscillator ol the cathode- coupled Butler type. A selected harmonic of the oscillator is mixed with the frequency of the vari- able frequency oscillator in the 1st mixer. A dual triode tube, V401, is employed as the oscillator tube, which has two output connections. Section A is tuned to the 5th harmonic of the crystal and feeds the grid of frequency multiplier V403B. The plate output of the grounded-grid oscillator section B is coupled through capacitor C407 to the harmonic generator grid. The crystal unit, which is of the type CR-28/U, is mounted in a type HD-54/U crystal oven. The oven employs two heaters and thermostats, one heater being used to bring the temperature quickly up to the operating level, whereas the other, which has a lower wattage, is used to maintain constant oper- ating temperature. In order to check the oscillator for proper operation, a test connection for meas- uring rectified grid current is brought out to test socket X412. The stability of the final transmitter frequency is more dependent upon the stability of this oscillator than upon that of any of the others. The reason is that the 1st crystal oscillator controls a greater percentage of the final fre- quency, especially in the high band, than do the other oscillators. This can readily be seen if we visualize the final frequency as being synthesized by adding together the crystal-oscillator frequen- cies in the simplified block diagram of figure 1-193. The key function of this oscillator is the reason why the highly stable Butler circuit is employed. 1-473. HARMONIC GENERATOR. The function of harmonic generator V403A is to produce any selected harmonic of the first crystal oscillator WADC TR 56-156 301 Section I Crystal Oscillators WADC TR 56-156 302 Figuro 1-194. Frequency diagram of Frequency-confro/ sysfem in Radio Set AN/ARC-33 Sech'on I Crystal Oscillators Figure I-f95. Monitor chassis. Block diagram of a~f-e system in Radio Set AN/AftC-33 output from the 4th to the 13th, inclusive. The selected harmonic is fed to the 1st mixer to be heterodyned with the fi-equency from the vari- able frequency oscillator. Harmonic selection is achieved by capacitance tuning the primary and secondary of the harmonic generator output transformer T401 to the desired harmonic fre- quency. The switching is accomplished by 10- position rotary switches S403A and S403B, which are driven by the selector motor through a harmonic generator clutch. These switches merely select the proper fixed capacitors for tun- ing the primary and scKiondary winding to the desired harmonic of the' 833.333-kc fundamental. 1-474. R-f input from the 1st crystal oscillator is fed to the harmonic generator tube, V403A, thi-ough coupling capacitor C407. A grid bias far below cutoff is piovided by grid resistor R407 in order to ensure an output rich in harmonics. The primary and secondary windings of transformer T401 are permeability tuned for alignment at the lowest (4th) harmonic. The highest frequency, which is the 13th harmonic, is determined by capacitoi s C418 and C419 across the primary and C42fl and C421 across the .secondary. Capacitors C418 and C421 are trimmers for alignment at the highest frequency. The .selection of all har- monics, up to but not including the 13th, is accomplished by switching in the proper fixed capacitor C U)8 through C417 across the primary of ti-ansfoimer TlOl, and C222 through C431 across the secondary. For the lowest harmonic, C416 and Cl 17 for the primary and C430 and C431 for the secondary are connected in parallel. For the highest harmonic, no auxiliarv capacitor is switched into the circuit. For proper operation of the monitor, it is necessary that the input level of the 1st mixer be approximately the same for each selected harmonic. This is accomplished by selecting a grid bias for the harmonic generator WADC TR 56-156 303 Section I Crystal Oscillators which causes the output level to vary inversely with frequency under a constant plate load im- pedance. Then, by maintaining the “Q” of the transformer T401 windings constant at all se- lected frequencies, the resultant plate load im- pedance of tube V403 can be made to vary directly with frequency because of the increasing induc- tance/capacitance ratio. The resultant voltage across the T401 transformer windings, therefore, is essentially constant regardless of the harmonic selected. 1-475. FIRST MIXER. The 1st mixer tube, V402, combines the output of the harmonic generator with the output frequency of the variable fre- quency oscillator (vfo) whose frequency is to be controlled by the monitor circuits. The 1st mixer output consists of the difference frequency be- tween the selected harmonic generator frequency and the v-f-o frequency, the latter frc^jnency al- ways being the higher. Input from the harmonic generator is fed directly to the control grid, whereas the input from the vfo, which enters the monitor chassis through connector J401, is fed from isolation amplifier V402A to the cathode of the 1st mixer through coupling capacitor C432. Output coupling is provided by bandpass coil assembly Z401, which consists of a low-band cir- cuit and a high-band circuit. The bandpass selec- tor relay selects the proper circuit of bandpass coil assembly Z401. The reason for using two bandpass circuits is that the broad frequency range of the radio set makes it necessary to divide the range into two smaller ranges. The band- determining factor is the selection of the 1st digit of the desired channel frequency at the con- trol panel. The control pane! switch energizes the band selector relay when the 1st digit is 2; that is, when the antenna frequency is to be less than 300 me. For antenna frequencies of 300 me and above, the band relay is unenergized. Coil assembly Z401 is designed to pass a band from approximately 12.1 to 12.9 me and a band from approximately 20.4 to 21.2 me. Because of the use of two bandpasses at the 1st mixer output, each of the harmonic generator selected output fre- quencies is used twice, once in each band. 1-476. BANDPASS AMPLIFIER. The bandpass amplifier, V404, is an amplifier for the 1st mixer output on the low-frequency band and is used to improve the bandpass characteristics of the cir- cuit. This stage is unconventional inasmuch as it also functions as a 2nd mixer, the operation of which is described in the following paragraph. The plate load consists of transformer T402, which is tuned to the same frequency band as the low band of Z401. The bandpass amplifier is capacitance-coupled through capacitor C443 to the number one grid of the 3rd mixer, V405. 1-477. SECOND MIXER. The 2nd mixer stage utilizes the same tube elem.ents of V404 as are used when the tube is operated as a bandpass amplifier. However, this dual usage does not occur simultaneously. Tube V404 functions as a 2nd mixer only when the radio set operates in the high band of the frequency range. The 2nd mixer combines the 1st mixer output frequency with the 10th harmonic of the 1st crystal oscillator fre- quency. The output consists of the difference frequency, where the 10th harmonic frequency is always lower than the output frequency of the 1st mixer. Note that the reduction of the 1st mixer output frequency by an amount equal to the 10th harmonic of the 1st crystal oscillator frequency is equivalent to extending the harmonic generator range from the 13th to the 23rd har- monic and eliminating the high band of bandpass coil assembly Z401. Input to the 2nd mixer from the high-band bandpass filter is fed to the control grid along with the input from frequency multi- plier V403B. The plate load consists of trans- foi'mer T402, which is tuned to pass a band from 12.1 to 12.9 me. The desired band width for the transformer is obtained through the use of load- ing resistors R418 and R419. Transformer T402 is the same plate load circuit for the 2nd mixer as is used when the stage operates as a bandpass amplifier. 1-478. FREQUENCY MULTIPLIER. The fre- quency multiplier, V403B, is fed with the 5th harmonic output from the 1st crystal oscillator. The output transformer, T406, is tuned to 8.33333 me, which is the 2nd harmonic of the multiplier input frequency. Thus, V403B is a frequency doubler and feeds its output, the 10th harmonic of the 1st crystal oscillator frequency, to the control grid of the 2nd mixer, V404. When the radio set is operated in the low band of its fre- quency range, no output is derived from the fre- quency multiplier because plate voltage is removed from tube V403B. The application of plate voltage is controlled by the band selector relay. When the band selector relay switches the high bandpass filter into the 1st mixer circuit, it also applies voltage to the plate of the frequency multiplier so that an 8.33333-mc signal is fed to the 2nd mixer. The frequency multiplier is capacitively coupled to the control grid of the 2nd mixer through capacitor C457. 1-479. SECOND CRYSTAL OSCILLATOR. The 2nd ci'ystal oscillator provides a selection of any WAOC TR 56-156 304 WADC TR 56-156 Section I Crystal Oscillators 4TH MIXER JAN 572S/6ASEV V407 X407 DISCRIMINATOR JAN 57E6/ 6N.5W V409 TEST SOCKET X4I2 (TOP VIEW) PIN I aUTCN CONTROl VOITAGE 2 +200V TEST 3 +I05V REG TEST 4 1ST OSC TEST 5 2 NO OSC TEST 6 3R0 OSC TEST T 4TH OSC TEST 8 GROUND 9 DISC OUTPUT K) GROUND II DISC TEST 0 409 (E3400M^ (4 3233333 ) (D3 566667 ) (D3 900000 ') (C3.733333 ) MONITOR OVENS AN/ARC'33 CRYSTAL PLACEMENT 0410 (S 5. 181250 ) C F3.650000() (R5.I729I7 ) ( C3733333 ) (0 3 750000 ) (h not used ) CP3 666667 ) (JS 716667 ) (03 766667 ) (K3 800000) (N3.683333 ) (L 3 700000) (M3 783333 ) csobJ, TOOOll \ | 7000 [/a^/^ _C5I0 2000 C5lll 2000 \ ■ HES3 ■ Mi'MI Tc5I4 -4- J 7000 UNDERSIDE VIEW OF SEALED RELAYS d) ® C®®® CD (j!)® (iD @ I note: ALL RESISTANCE IN OHMS ALL CAPACITANCE IN MMF UNLESS OTHERWISE STATED. X^IOOO COLOR a E!] BLACK m Miai a GREEN n HIM a RED m SLATE a VIOLET □1 WHITE ai YELLOW S3 Figure 1-196. Monitor chassis. Schematic diagram of a-f-c system in Radio Set AN/ARC-33 o 305-306 one of five crystal-controlled frequencies by means of crystal switching. The crystal-controlled out- put frequency is' fed to the suppressor grid of the 3rd mixer to be mixed with either the 2nd mixer or the bandpass amplifier output frequency. The oscillator circuit utilizes one half of the duplex triode tube V406 as a grounded-plate Pierce oscillator. R-f output is taken from the cathode. Each of the five crystal units, which are of the CR-27/U type, has a separate trimmer so that each selected 2 nd crystal oscillator frequency can be adjusted exactly. All of the crystal units with their trimmers are enclosed in a single oven (Bendix Radio type L205628) which is kept at approximately 75°C (167°F) by a thermostat- controlled heater. A booster heater and thermo- stat are provided in addition for quick warmup. In order that the oscillator operation may be checked, a test connection from the grid circuit is brought out for checking rectified grid current at test socket X412. 1-480. THIRD MIXER. The circuit of the 3rd mixer, V405, is similar to that of the 2nd mixer except that the signal voltage from its heterodyne crystal oscillator is injected at the suppressor grid. The 3rd mixer output circuit is tuned to a center frequency of 8.9 me and is designed to pass a band approximately from 8.8 to 9.0 me. The 3rd mixer combines the output frequencies of either the 2 nd mixer or bandpass amplifier and the 2 nd crystal oscillator, the output frequency of the former two always being the higher. The 3rd mixer output, which is the difference frequency, is fed from the secondary of transformer T403 to the 4th mixer through capacitor C449. 1-481. THIRD CRYSTAL OSCILLATOR. The 3rd crystal oscillator, the V406B circuit, is of the same design as the 2 nd crystal oscillator except for the operating frequencies. The crystal units, which are of the CR-27/U type, are divided into two groups of five each. All are used throughout the total frequency range of the radio set. The 3rd crystal oscillator employs the second half of the same tube that is used for the 2 nd crystal oscillator. The 10 crystal units with their trim- mers are housed in a 13-position oven (Bendix Radio type N205651), of which one position is not used. The other two positions are used to mount the two crystals of the 4th crystal oscil- lator. The oven is thermostatically controlled at 75°C (167°F). A separate booster heater and thermostat are provided for quick warmup. 1-482. FOURTH MIXER. The 4th mixer, V407, is identical to the 3rd mixer and operates in the same manner. The 4th mixer output circuit is Section I Crystal Oscillators designed to pass a band of approximately 5.1 to 5.2 me. In the 4th mixer are combined the output frequency of the 3rd mixer and that of the 3rd crystal oscillator, the former frequency always being the higher. The difference frequency is selected by the tuned plate transformer and is inductively coupled into the discriminator circuit. 1-483. DISCRIMINATOR. The purpose of the discriminator, V409, is to indicate a deviation in phase or frequency between its two inputs. One of these inputs is obtained from the 4th crystal oscillator and is used as the reference signal. The other discriminator input is derived from the 4th mixer output, and it is the deviation in phase of this signal from that of the reference signal that is to be indicated by means of the output voltage across R430 and R431. The magnitude of this discriminator voltage indicates the extent of the deviation, whereas the polarity indicates the di- rection of the deviation. The circuit design is similar to the type employed for frequency- modulation (FM) receivers, except that in this case the reference voltage is obtained from a separate reference oscillator instead of a parallel resonant tank circuit. The voltages applied to the discriminator are shown in figure 1-197. E, is the input from the 4th mixer and Ek is the reference voltage from the 4th oscillator. To simplify the discriminator explanation, Er, insofar as it adds vectorially with Ee to form the r-f voltages across the two diodes, is best interpreted in terms of two equal and separate voltages 180 degrees out of phase. One is E;;n, the r-f voltage at pin 2 with respect to point B; the other is Etei, the r-f volt- age at pin 7 with respect to point B. Assuming that the bypass capacitors C452 and C453 offer zero impedances to the r-f signals, pins 1 and 5 and point A are all at r-f ground potential. Thus, the r-f voltage (E:;) of plate pin 2 with respect to cathode pin 5 is the same as the voltage of pin 2 with respect to point A. Similarly, E 7 , the r-f plate voltage of the second diode, is equal to the voltage of pin 7 with respect to point A. Now, Ei and E 7 have one voltage component in common, which is El, the voltage of point B with respect to point A. Thus, Ea and E 7 are equal to the re- sultants, respectively, obtained by adding vector- ially to the common voltage E, the voltages E^ei and E 7 E 1 . If we assume that C452 charges to the peak of Ea and that C453 charges to the peak of E7, the d-c polarities will be as indicated in figure 1-197 (D), where point A is shown as negative with respect to the two cathodes. The d-c output equals (Ea — E7), where both voltage symbols represent the positive peak magnitudes only. The WADC TR 56-156 307 Section I Crystal Oscillators output is positive or negative according to whether Ea is greater or less than E 7 , respec- tively; and this is dependent, respectively, upon whether E, lags or leads the zero-output position, which is the 90-degree phase displacement from E„ that is shown in figure 1-197 (A). Upon exam- ination of the discriminator diagram it can be seen that the application of either of the input voltages alone does not develop a discriminator output, since in each case E 2 and E 7 will equal each other, and hence equal currents will flow in opposite directions through the two halves of dis- criminator load R430 and R431. Also, it can be seen from the vector diagram (A) that, if the two discriminator input voltages are exactly 90 degrees out of phase, the resultant voltages ap- plied to the diode plates are equal, thus pi'oducing zero discriminatoi- output. If, however, the vari- able frequency oscillator should vary in phase, the resultant voltages applied to the discriminator plates are unequal, as shown in vector diagrams (B) and (C). This results in a discriminator out- put voltage whose value and polarity depend upon the magnitude and direction, respectively, of the phase deviation. If there is a frequency deviation, the phase of the 4th mixer output, E,, rotates completely around the 4th crystal oscillator out- put, Er, with a consequent a-c voltage appearing in the discriminator output. The frequency of this ac is equal to the difference between the two discriminator input frequencies. 1-484. Discriminator output is applied to the tuning-motor-control amplifier and to the reac- PIN 7 FOURTH MIXER OUTPUT E] 90' OUT OF PHASE WITH FOURTH CRYSTAL VOLTAGE Er- DISCRIMINATOR OUTPUT* ZERO |MAX E 2 I-IMAX E 7 I* ZERO D-C OUTPUT. A PIN 2 (R-F GROUND) (A) PIN fourth mixer OUTPUT E] LAGGING NORMAL PHASE. INDICATING THAT v-F -0 frequency is TENDING TO GO LOW. RESULTING POSITIVE DISCRIMINATOR OUTPUT WILL CAUSE MOTOR TO TUNE VFO HIGHER IN FREQUENCY. |MAX E 2 I -|MAX E 7 I* POSITIVE 0-C OUTPUT. A (R-F GROUND) (B) PIN 7 FOURTH MIXER OUTPUT E| LEADING NORMAL PHASE, INDICATING THAT V-F-O FREQUENCY IS TENDING TO GO HIGH. RESULTING NEGATIVE DISCRIMINATOR OUTPUT WILL CAUSE MOTOR TO TUNE VFO LOWER IN FREQUENCY. }MAX E 2 I - (MAX E 7 I • NEGATIVE 0-C OUTPUT. GROUND) OUTPUT TO REACTANCE TUBE AND MOTOR DISCRIMINATOR CONTROL AMPLIFIER (D) Ej* INPUT FROM 4TH MIXER Er» REFERENCE VOLTAGE FROM FOURTH CRYSTAL OSOLLATOR • ±(^28 + E7b)“±(E2B “£ 73 ) Eg -RESULTANT VOLTAGE APPLIED TO DIODE PLATE (SOCKET PIN 2) E 7 -RESULTANT VOLTAGE APPLIED TO DIODE PLATE (SOCKET PIN 7) K- IN ACTUAL PRACTICE THE V-F-O FREQUENCY CONTROL CIRCUITS ARE SUCH THAT "ON FREQUENCY" DISCRIMINATOR OUTPUT IS APPROX +5 VOLTS. THUS. IN NORMAL OPERATION, VECTOR Ej WILL BE TILTED SLIGHTLY TO THE RIGHT OF ITS 90* POSITION SHOWN IN FIGURE (A) AT LEFT. DUE TO THE ACTION OF THE DIODES AND THEIR LOADS. THE DISCRIMINATOR RESULTANT 0-C OUTPUT IS EQUAL TO THE DIFFERENCE IN MAGNITUDE BETWEEN Eg AND E 7 ({MAX Egl-jMAX E 7 I). (C) Figure 1-197. Discriminafor operation as a function of the phase difference be/ween the input voltages WADC TR 56-156 308 Section I Crystal Oscillators tance tube shunted across the variable frequency oscillator. When the variable frequency oscillator approaches the desired operating frequency, a point is reached when the discriminator output frequency becomes equal to or less than the maxi- mum frequency at which the reactance tube can respond. At this point the reactance tube imme- diately locks the variable frequency oscillator exactly to its correct frequency. When this occurs, the discriminator output is dc and is proportional to the phase difference between its two input voltages, which difference is, in turn, proportional to the amount of “pull” exerted by the reactance tube. This d-c control voltage is fed to the motor- control amplifier, V411, and causes the tuning motor to drive the variable frequency oscillator tuning capacitor to that point which eliminates excessive pull by the reactance tube. The circuit design is such that the equilibrium point corre- sponds to a discriminator output of approximately 5 volts positive. Normally it would be assumed that the “on-frequency” point would be reached when the discriminator output dropped to zero, which would occur when the two input signals were exactly 90 degrees out of phase. The zero- voltage state is not used, however, in order to avoid ambiguity in identifying the on-frequency condition and to simplify the control circuit by having an equilibrium control voltage of definite magnitude. For example, zero output not only occurs at each of the two (plus and minus) 90- degree phase conditions, but also occurs when the variable frequency oscillator is so far off fre- quency that there is no input from the 4th mixer, or if one or both of the input voltages fails, or when the two input frequencies are different and the discriminator records the best frequency. Thus, a plus 5-volt reference is used, and the control circuits are so biased that this control level establishes the on-frequency condition. The ac which the discriminator develops before the reactance tube “pull-in” point is reached, is pre- vented from affecting the motor-control amplifier by the low-pass resistance-capacitance filter, R451 and C445A. 1-485. By employing a phase-sensitive discrim- inator, it is possible to feed a correction voltage to the control circuits before an actual frequency deviation occurs, since a frequency deviation, un- less it is an instantaneous, discontinuous jump, is first indicated as a phase deviation. Therefore, no frequency error occurs except that which may be due to the reference crystals themselves. All small and rapid frequency shifts of the variable frequency oscillator are corrected by the reactance tube. Larger and slower drifts are corrected by the motor-control amplifier and the tuning motor. An extremely large and sudden frequency jump of the variable frequency oscillator which takes it out of the range of the reactance tube causes the entire tuning sequence to recycle. However, this does not occur during normal operation. To aid in discriminator alignment and test, two test points are brought out to pins of test socket X412. One of these, pin 9, makes it possible to measure the total discriminator output. The other, pin 11, is connected to the load center-tap for checking the discriminator operation. 1-486. FOURTH CRYSTAL OSCILLATOR. The 4th crystal oscillator is used to control the oper- ating frequency of the discriminator and thereby has final control of the exact frequency of the variable frequency oscillator. The 4th crystal oscillator can be switched to either one of two type CR-27/U crystal units, whose frequencies are spaced 8.333 kc apart. This spacing is equal to one-twelfth of the channel spacing of 100 kc. Thus, it is the spacing of the 4th crystal oscillator frequencies that determines the channel spacing. For maximum stability the crystal units are mounted in the same crystal oven that houses the crystals of the 3rd crystal oscillator. Crystal selection is controlled by a selector switch on the main control panel. The selection is determined by the choice of the 4th digit of the channel frequency. 1-487. The 4th crystal oscillator circuit employs one half of a duplex triode tube, V410, which is connected as a radio-frequency grounded-plate Pierce oscillator, with the circuit designed in a manner similar to those of the 2nd and 3rd crystal oscillators. The output is taken from the cathode and coupled through C471 to the grid of the other half of tube V410, which is operated as a cathode follower. The output of the cathode follower is inductively coupled into the discriminator circuit through transformer T405. The purpose of the cathode follower is to isolate the loading effects of the discriminator from the 4th crystal oscil- lator. As a check on the 4th crystal oscillator operation, a test connection for rectified grid cur- rent measurement is brought out to test socket X412, pin 7. 1-488. CLUTCH CONTROL TUBE. The purpose of the clutch control tube V408, is to shift the r-f head tuning drive from the medium-speed clutch to the low-speed clutch for fine tuning of the exact channel frequency. Whenever the fre- quency of the variable frequency oscillator, after WADC TR 56-156 309 Section I Crystal Oscillators being fed through the four mixers, comes within range of the discriminator input tuning at trans- former T404, the tuning should be shifted into low speed. Thus, by coupling part of the discrim- inator input through capacitor C464 to clutch- control rectifier V408A and amplifying the recti- fier output through V408B, a control voltage is obtained which is used ito operate the clutch- control relay. The clutch-control relay is mounted on the main frame and has its coil connected in series with the plate of clutch-control amplifier V408B. The relay is energized when the tube grid receives no excitation. When a signal is fed to the discriminator from the 4th mixer, a portion of the signal is fed through capacitor C464 and rectified in diode-connected tube V408A. The re- sulting current flow in resistor R443 causes the ungrounded end to become more negative. This negative voltage is fed to the grid of the clutch- control amplifier through low-pass filter R446 and C463, cutting the tube off. This in turn de-energizes the clutch-control relay and places the low-speed clutch in operation. It should be noted here that the over-all tuning range of the radio set is divided into two nearly equal bands and that while the variable frequency oscillator tuning is being driven through the unused band a spurious frequency may pass through to the discriminator and the clutch-control tube. In this case, however, even though the clutch-control relay is de-energized, the tuning remains in high speed because of a lockup circuit. 1-489. MOTOR-CONTROL AMPLIFIER. The motor-control amplifier, V411, is a d-c amplifier whose purpose is the control of a reversible tun- ing motor in accordance with a d-c control voltage from the discriminator. The discriminator output voltage is fed to the grid through the low-pass resistance-capacitance filter R451 and C445A so that the ac, which appears in the discriminator output as the channel frequency is being tuned, does not influence the operation of the amplifier. As its plate load, the motor-control amplifier works into two relay coils in series. These are called the high and low discriminator relays. The two relays are mounted on the main frame. The high discriminator relay is designed to “pull in” - . transmitter output v-f-o frequency = The screen grid of the v-f-o tube, V802, is the anode of the oscillatory circuit. The plate circuit is tuned to three times the grid frequency, thereby tripling in this tube. Coupling capacitor C907 from grid inductor 1801 feeds a small amount of at approximately 9 ma ^nd to “drop out” at ap- proximately 6 ma. The low discriminator relay is designed to pull in at approximately 5 ma and to drop out at approximately 3 ma. The plate current of the motor-control amplifier is so adjusted that, with a plus 5-volt “on-frequency” output from the discriminator, approximately 5.5 ma plate cur- rent flows. This is sufficient to energize the low discriminator relay but not the high one. The contacts of the low discriminator relay are so con- nected that when the relay is energized the tuning motor (mounted in the r-f head) is de-energized. If the frequency of the variable frequency oscil- lator should tend to drift too low, the discrimina- tor voltage controlling the amplifier becomes more positive, increasing the amplifier plate current and thereby energizing both relays. This has the effect of causing the tuning motor to turn in a direction that diminishes the tuning capacitance and hence raises the frequency. If the variable frequency should tend to drift too high, the plate current of the motor-control amplifier decreases to a value below 3 ma and both relays are de- energized. This has the effect of causing the tuning motor to turn in the opposite direction, so that the frequency of the variable frequency oscil- lator is decreased. Note that in order to keep the motor-control amplifier bias constant, regardless of plate current flow, a positive 6-volt cathode bias is obtained from the d-c drop across the heater, rather than from the drop across a cath- ode biasing resistor. R-F Head 1-490. The r-f head contains all the main channel transmitter and receiver r-f circuits including the reactance tube, the tuning capacitor drive mech- anism, and certain tuning control circuits. A block diagram is shown in figure 1-198 and a schematic diagram is shown in figure 1-199. Beginning with the variable frequency oscillator, we shall discuss first the transmitter circuits and then the re- ceiver circuits. 1-491. VARIABLE FREQUENCY OSCILLATOR AND TRIPLER. The variable frequency oscilla- tor employs an electron-coupled Hartley circuit. The v-f-o frequency is given by the formula: freq — receiver 1st intermediate freq 12 r-f energy to the frequency-control circuits in the monitor. A reactance tube is shunted across the oscillator tank for fine frequency control. Oscilla- tor-tripler output is capacitively coupled to the untuned grid of the doubler tube, V803. WADC 7R 56-156 310 Section I Crystal Oscillators TO couuti TO uc fm n coaum. HEUTOOOt -KUDWOIO lEUtT 0602 OOtUMOKC CHASSIS OHOUAWIIEC nSCIIMUTK ISOLATIW NOTOH CWTOOI. SUTPOIOT AUDIO CHASSIS H AID AUDIO AMPIIFIEI IELATS NODULATOfI CHASSIS Figure 1-198. K-f head. Block diagram of main channel transmiHer and receiver, r-f circuits in Radio Set AN/ARC-33 1-492. REACTANCE TUBE. The reactance tube, V801, is a device that represents a variable and controllable reactance shunted across the tuned grid circuit of the variable frequency oscillator. Its purpose is to convert d-c control voltage from the monitor into a small frequency variation of the variable frequency oscillator. Radio-frequency voltages from the oscillator are coupled through capacitor C805 to the plate of the reactance tube and directly to a network made up of capacitors C804 and C818, resistors R801 and R804, and the grid-cathode capacitance of the tube, C^. Capacitor C804 is a blocking capacitor, so that as far as the a-c functioning of the tube is con- cerned, it need not be considered. Cathode bypass capacitor C803 is sufficiently large for the cathode to be considered at r-f ground potential. Resistor R802, inductor L829, and capacitors C801 and C802 provide a de-coupling network and filter through which the d-c control voltage is applied to the control grid without shunting its r-f input impedance. Resistor R807 is the plate feed com- ponent. Resistors R801 and R804 and capacitor C818 in conjunction with the grid-cathode capaci- tance, Cc, form a phase-shifting network such that the r-f voltage applied to the grid lags the oscil- lator voltage input to the network by 90 degrees approximately. Since the plate current of react- ance tube V801 is in phase with the grid voltage, the plate current flow lags the oscillator r-f volt- age applied to the plate by approximately 90 degrees. This appears to the variable frequency oscillator as an inductive reactance since the cur- rent in an inductor lags the applied voltage by 90 degrees. By controlling the d-c bias applied to the grid, it is possible to control the amplitude of the current, and thus the magnitude of the effec- tive inductive reactance shunting the oscillator tank. In this way, the frequency of the variable frequency oscillator can be controlled within a narrow range by means of a d-c control voltage. A positive control voltage increases the reactive current in the tube and thereby decreases the effective inductive reactance across the oscillator tank. The effect, therefore, is to cause an increase in the frequency. A negative-going control voltage has the opposite effect. The reactance tube is biased by cathode resistor R805 by an amount WADC TR 56-156 311 Section I Crystal Oscillators that permits normal “onrfrequency” oscillator operation when the control voltage from the mon- itor is approximately plus 5 volts. 1-493. FIRST DOUBLER. The first doubler of the r-f head is a conventional grid-leak-biased frequency multiplier. The stage employs tube V803 with an output circuit tuned to twice the input frequency. The plate circuit consists of two inductors in parallel in order to increase their sizes and make it possible to employ coiled induc- tors rather than a more space-consuming linear line. The tuning is ganged with the other r-f tuned circuits as shown in the schematic diagram. Radio-frequency energy used for heterodyne oscil- lator injection into the main channel receiver 1st mixer is taken off at a tap of plate tank L802. The variable frequency oscillator-tripler, the re- actance tube, and the doubler are continuously supplied with plate and screen voltages since these circuits are employed in both transmit and receive operation. From the latter stage to the antenna, however, plate and screen voltages are removed from the transmitter tubes during re- ceive operation. 1-494. TRANSMITTER MIXER. It is in the mixer that a 7.6625-mc signal from the sidestep oscillator is added to the first doubler output to make the mixer output frequency exactly one half of the transmitting antenna frequency. (Remem- ber that 7.6625 me is one half of the 1st receiver intermediate frequency of 15.325 me.) A conven- tional mixer circuit employing tube V805 is used here, with the side-step oscillator voltage being injected at the control grid through capacitor C821 and with the doubler output being fed to the control grid through capacitor C820. The mixer plate circuit is tuned to the sum frequency by means of a doubler-inductor arrangement. One of the inductors is wound with concentric cable, which allows the tuning capacitor rotor and the cold ends of the inductors to be returned to ground. The center conductor is connected to the plate on the hot end and is bypassed to ground on the cold end by capacitor C825. Plate voltage is supplied to tube V805 through resistor R819. 1-495. SIDESTEP CRYSTAL OSCILLATOR. The sidestep crystal oscillator employs pentode V804, connected as a triode and operating on a single frequency of 7.6625 me. The circuit is arranged as a Pierce oscillator with the crystal connected directly from the plate to the grid of the oscillator tube. A type CR-18/U crystal unit is employed. Since this oscillator controls only a small percentage of the final transmitter fre- quency, its normal operating stability is sufficient WADC TR 56-156 without the use of thermostatically controlled temperature for the crystal unit. Note that an 18-/vif capacitor is shunted directly across the crystal unit and that no externally connected plate-to-ground capacitor is used. The 18 ^ rep- resents 60 per cent of the required load capaci- tance for the type CR-18/U crystal unit. With the grid-to-ground capacitance probably 8 or 10 greater than the 68 n4 of the externally con- nected grid-ground capacitor C822, it appears that the circuit has been designed to provide a maximum output voltage consistent with the drive-level and load-capacitance requirements of the crystal unit, rather than for maximum fre- quency stability. The oscillator is fed plate voltage through the voltage-dropping resistor R815. The oscillator output is capacitively coupled from the plate of the oscillator tube to the control grid of mixer tube V805 through C821. Plate voltage is derived from the plus 200-volt transmitter sup- ply, so that the tube operates only when the radio set is turned to the transmit position. 1-496. AMPLIFIER. This is a conventional grid- leak-biased r-f amplifier employing pentode V806. The circuit acts as a direct amplifier with the output circuit tuned to the same frequency as the input circuit. The input is capacitance-coupled to the preceding mixer stage through capacitor C828 and the output is capacitance-coupled to doubler V807 through capacitor C832. The amplifier tuned output circuit is a double-inductor tank of the same type that is used in the plate circuit of the transmitter mixer. 1-497. TRANSMITTER DOUBLER. The doubler, V807, is a grid-leak-biased stage that employs a cavity-tuned plate circuit tuned to twice the input frequency. A pencil triode tube is used because its small size permits superior u-h-f operating characteristics. 1-498. INTERMEDIATE POWER AMPLIFIER. The intermediate power amplifier (IPA) employs a lighthouse tube, V808, in a grounded-grid cir- cuit with cavity tuning of the plate. The cavity is tuned to the doubler output frequency, providing an IPA output frequency equal to the transmitter antenna frequency. Radio-frequency input is coupled magnetically from the output of the pre- ceding doubler tube, V807, through coupling loop L809, which applies rf between ground and the IPA cathode through capacitor C838. Concentric inductor L810 is inserted in the heater circuit to make it at the same r-f potential as the cathode. Capacitor C838 also blo<^s the 12.6-volt d-c fila- ment potential from shorting to ground through the ground coupling loop L809. One heater con- 312 WADC TR 56-156 +400Vi AUDIO MODULATOR OUTPUT Figure I -199. R-f head. Scheme transmitter and receiver r-f ctrci J.-. Scdton i Crystal Osdflators FlUlNfUFO 1 m 2C39* Figure 1-199. K-f head. Schematic diagram of main channel transmitter and receiver r-f circuits in Radio Set AN/ARC-33 c 313-314 nection and the cathode connection are common. The heater circuit is wired in series with the heater of the final amplifier so as to provide the required 12.6-volt total drop across the entire circuit after passing through dropping resistor R833. A voltage divider composed of resistors R830 and R832 is connected between one side of the filament and ground. The grid return is con- nected to the tap to provide the proper negative bias voltage. These resistors are high in value so that they do not influence the heater voltage and current. IPA output is coupled to the final power amplifier input through an inductive coupling loop, L812, in the cavity. Since the final power amplifier is a grounded-grid r-f amplifier, it can- not be plate modulated 100 per cent unless the output of the exciting stage also is modulated. Therefore, to be able to modulate fully the final amplifier output, the plates of doubler V807 and IPA V808 are modulated by the same audio volt- age as the power amplifier. 1-499. FINAL POWER AMPLIFIER. The final r-f power amplifier, V809, is similar in design and operation to the intermediate power amplifier. Grid bias for the final amplifier is derived from resistor R835. Resistor R834 supplies a cathode bias. Radio-frequency output to the antenna is coupled inductively to the cavity magnetic field by pickup loop L815 and is fed to the antenna receive-transmit changeover relay through a co- axial cable. The coaxial connector is mounted directly on the cavity assembly, 0802. To provide a sidetone voltage that is indicative of the trans- mission quality, the sidetone voltage is obtained from the final r-f output cavity through inductive coupling L814. This r-f voltage is rectified through the sidetone crystal rectifier, CR801, and applied to the i-f and audio chassis to be fed through the receiver audio system to the operator’s headset. 1-500. MAIN CHANNEL RECEIVER R-F CIR- CUITS. The main channel receiver includes two r-f amplifier stages, a mixer, a frequency doubler, and a 15.325-mc i-f amplifier. Both r-f amplifiers and the doubler are cavity-tuned. The doubler is included to multiply the output frequency of the first r-f-head doubler. By employing a heterodyne injection frequency controlled by the same vari- able frequency oscillator that controls the trans- mitter frequency, perfect tracking between the transmitter and receiver tuning is possible. The i-f output is fed by coaxial cable to the second mixer and the 2.8-mc intermediate frequency am- plifier on the i-f and audio chassis. 1-501. FIRST R-F AMPLIFIER. The first r-f amplifier, V814, is a grounded-grid cavity-tuned Section I Crystal Oscillators amplifier with a shunt-fed plate. The cavity is similar to the cavities used in the transmitter sec- tion and operates in exactly the same way except that the amplifier tube is located outside of the cavity. Electrical connections are made through holes in the cavity walls. Tube V814 is tapped down on the cavity center column to prevent load- ing of the tuned circuit by the tube. The antenna input is also tapped down to match the character- istic impedance of the coaxial line. The grounded- grid circuit is effective in preventing coupling be- tween the input and output circuits, thus prevent- ing oscillations due to feedback through the tube capacitance. Also, the grounded-grid design with the cathode-injection input provides a more c?»n- stant input impedance over the frequency range. Two plate and two grid connections are provided on the JAN-6F4 tubes to make it possible to use shorter leads and to obtain better bypassing bal- ance, as is illustrated by the V814 grid connections to ground through capacitors C892, C893, C898, and C899. Examination of the tuned-cavity cir- cuits will show that only the first input cavity employs a capacitance connection, represented by C896. Since this cavity is not loaded by the plate capacitance of a vacuum tube, capacitor C896 is added in order for all three cavities to have sim- ilar tuning characteristics. Automatic-ga